Network Resilience

Network resilience

Xueming Liua, Daqing Lib, Manqing Mac, Boleslaw K. Szymanskid, H Eugene Stanleye, Jianxi Gaof

aKey Laboratory of Image Information Processing and Intelligent Control, School of Artiﬁcial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China bSchool of Reliability and Systems Engineering, Beihang University, Beijing 100191, China cDepartment of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180; Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, NY 12180 dDepartment of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180; Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, NY 12180 eCenter for Polymer Studies, Department of Physics, Boston University, Boston, MA 02215; fDepartment of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180; Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, NY 12180 (e-mail: [email protected])

Abstract Many systems on our planet are known to shift abruptly and irreversibly from one state to another when they are forced across a “tipping point,” such as mass extinctions in ecological networks, cascading failures in infrastructure systems, and social convention changes in human and animal networks. Such a regime shift demonstrates a system’s resilience that characterizes the ability of a system to adjust its activity to retain its basic functionality in the face of internal disturbances or external environmental changes. In the past 50 years, attention was almost exclusively given to low dimensional systems and calibration of their resilience functions and indicators of early warning signals without considerations for the interactions between the components. Only in recent years, taking advantages of the network theory and lavish real data sets, network scientists have directed their interest to the real-world complex networked multidimensional systems and their resilience function and early warning indicators. This report is devoted to a comprehensive review of resilience function and regime shift of complex systems in diﬀerent domains, such as ecology, biology, social systems and infrastructure. We cover the related research about empirical observations, experimental studies, mathematical modeling, and theoretical analysis. We also discuss some ambiguous deﬁnitions, such as robustness, resilience, and stability. Keywords: Complex networks; Resilience; Nonlinear dynamics; Alternative stable states; Tipping points; Phase transitions

Contents 2.2.2 Individual abrupt phase transitions in real ecosystems . . . . 15 1 Introduction2 2.2.3 Regime shifts in a connected world ...... 16 2 Tipping points in ecological networks5 2.3 Predicting the tipping points in ecosys- 2.1 Multiple stable states and resilience in tems ...... 19 ecosystems ...... 6 arXiv:2007.14464v1 [q-bio.QM] 26 Jul 2020 2.3.1 Data-driven approach for low 2.1.1 Mathematical models for multi- dimensional systems ...... 19 ple stable states ...... 6 2.1.2 Empirical examples of ecosys- 2.3.2 Model-driven approach for systems with multiple stable states9 tems with networked dynamics . 21 2.1.3 Ecological resilience and engi- 2.3.3 Control tipping points in neering resilience ...... 11 ecosystems ...... 25 2.2 State transitions in ecosystems . . . . . 13 2.2.1 Diﬀerentiation between phased 3 Phase transitions in biological networks 27 shift and multiple stable states . 13 3.1 Bistability in biological systems . . . . 27

Preprint submitted to Physics Reports July 30, 2020 3.1.1 Generators of biological bista- 5.1.4 Rich club aspects of CIs . . . . 70 bility ...... 28 5.2 Failure models of critical infrastructures 70 3.1.2 Mathematical models of bio- 5.2.1 Cascading failure model . . . . 70 logical bistability ...... 28 5.2.2 Critical transitions ...... 71 3.1.3 Empirical studies of biological 5.2.3 Interdependent networks . . . . 73 bistability ...... 31 5.3 Infrastructure resilience ...... 74 3.2 Resilience at multiple levels of biologi- 5.3.1 Evaluation ...... 74 cal network ...... 33 5.3.2 Prediction ...... 79 3.2.1 Resilience of genetic circuits in 5.3.3 Adaptation and control . . . . . 80 presence of molecular noises . . 33 3.2.2 Resilience of unicellular organ- 6 Resilience, robustness, and stability 82 isms under environmental stress 35 6.1 Resilience and network dynamics . . . 83 3.2.3 Potential landscapes of cellular 6.2 Stability in complex systems ...... 84 processes in complex multicel- 6.3 Robustness in networks ...... 85 lular organisms ...... 38 3.3 Indicators of resilience in biological 7 Conclusions and future perspectives 87 systems ...... 43 7.1 Conclusions ...... 87 3.3.1 Early warning signals before bi- 7.2 Future perspectives ...... 88 ological populations collapse . . 43 7.2.1 Uncovering network dynamics 3.3.2 Critical slowing down as early form data on system evolution . 88 warning signals for the onset of 7.2.2 Empirical studies of resilience . 88 disease ...... 44 7.2.3 Resilience of networks of net- 3.3.3 Dynamic network biomarkers works ...... 88 in the progression of complex 7.2.4 Resilience of networks with in- diseases based on gene expres- complete information ...... 89 sion data ...... 46 7.2.5 Controlling the resilience of networks ...... 89 4 Behavior transitions in animal and human networks 49 1. Introduction 4.1 Cultural resilience of social systems . . 51 4.1.1 Naming Game model ...... 52 The nature and the world in which we live are filled 4.1.2 Committed minorities . . . . . 53 with changes and crisises [1,2,3,4]. Examples are the 4.1.3 Agent interaction models . . . . 57 climate change [5], the global pandemic of the novel 4.2 Survive resilience in social animal sys- coronavirus (2019-nCoV) that are changing the world tems ...... 60 [6], the catastrophe in east Africa caused by the infesta- 4.2.1 Social foraging and nest building 60 tion by desert locusts [7], and the 2019-2020 bushfire in 4.2.2 Social animals response to stress 61 Australia that burned through some 10 million hectares 4.3 Resilience of the planetary social- of land, with thousands of people displaced, and count- ecological system ...... 62 less animals killed by the raging fires [8,9]. In addi- 4.3.1 Early warning signal in social- tion, these threats and crisises are not independent but ecological system ...... 63 related with one another. For example, the Australia 4.3.2 Regime shifts in social- bushfire and locust swarms are linked to the oscillations ecological systems ...... 65 of the Indian Ocean Dipole, which is one aspect of the 4.3.3 Difficulties in the studies of re- growing of the global climate change [10, 11]. How the silience in social networks . . . 67 nature or societies response to such threats and crisises is defined by their resilience, which characterizes the 5 Failures in critical infrastructure (CI) sys- ability of a system to adjust its activity to retain its ba- tems 68 sic functionality in the face of internal disturbances or 5.1 Structural properties of CIs ...... 68 external changes [12]. 5.1.1 Spatial aspects of CIs ...... 69 Resilience is a defining property that is universally 5.1.2 Small-world connectivity of CIs 69 presenting in perhaps all dynamical systems. Just like 5.1.3 Modularity of CIs ...... 70 the bird Phoenix from the ancient Greek mythology, 2 which was cyclically reborn of its own ashes, the re- multifaceted [16]. There are more than 70 definitions silience in nature makes people witness an incredible for resilience in scientific literatures [17]. Some of them renewal of forests. As shown in Fig.1, regeneration generally spam multiple disciplines and some are pro- of the Australian forests happens, despite the fires still posed for specific systems. For instances, Holling [4] blazing there. However, not every system has the same used the notion of “resilience” to characterize the de- resilience resulting in strong ability for self-renewal. gree to which a system can endure perturbations without Damaged by the same Australians fires, the water, elec- collapsing or being carried into some new and qualita- tricity network, the communication network, the trans- tively different state. Haimes [18] points out two es- port and supply chains have been hit hard and the slow sential elements of resilience: the ability to withstand recovery times highlight a lack of resiliency in these presence of errors, and after a perturbation recover to systems [8]. the original stable state. Bruneau et al. [19] conceptu- alizes seismic resilience as the ability of both physical and social systems to withstand earthquake-generated forces and demands and to cope with earthquake impacts through situation assessment, rapid response, and active recovery strategies [20]. Generally, these definitions vary between three extremes: system resilience (or called ecological resilience) that focus on the magnitude of the change or perturbation that a system can endure without shifting into another stable state [4], engineering resilience, which is defined by the recovery rate [21, 22], and adaptive resilience that characterizes the Figure 1: Regrowth during the 2019-2020 Australia bushfire. capacity of social-ecological systems to adapt or trans- Figure from [9]. form in response to unfamiliar, unexpected and extreme shocks [23, 24]. Most definitions are trying to achieve a Depending on the forms and strengths of perturba- balance between these three flavors of resilience. tions, and the inherent resilience [4, 12], dynamical sys- During the past five decades, it has been a term in- tems may exhibit different responses to various pertur- creasingly employed throughout science and engineer- bations. (1) The system may maintain its original state ing, which makes resilience a multidisciplinary con- without any adverse effects. For example, in the Internet cept [16, 25]. Due to the often unknown intrinsic dy- network, routers choose the paths for sending packages namics in large scale systems and the limitation on based on the routing tables that keep track of how ef- analytic tools, most of the studies have been concen- fective any path through the network is [13]. If some trated on low-dimensional systems or time series data links fail due to malfunction or overload, then the In- analysis without modelling [26, 27, 28, 29, 30]. In ternet will dynamically reroute packages to maintain the studies with dynamical models, the resilience be- its functionality. (2) The system may lose some of its haviour is usually captured by a one-dimensional (or function but recovery after some time. For instance, the low-dimensional) nonlinear dynamic equation dx/dt = 2003 electricity blackout [14] affected much of Italy and f (β, x), where the functional form of f (β, x) shows the caused scenes of chaos: 110 trains were canceled, with system’s dynamics, and the parameter β represent the 30,000 people stranded on trains in the railway network, environmental conditions [31]. The rule of stability of and all flights in Italy were also canceled. However, af- motion [32] requires that f (β, x0) = 0 and ∂ f /∂x|x=x0 < ter several hours, electricity was restored gradually in 0, so one could get the resilience function x(β), which most places, and people’s daily life went back to nor- represents the possible states of the system as a func- mal. (3) The system may go to another attractor, and do tion of β. As shown in Fig.2, at some critical point βc, not come back to the original state. In ecology, climate the resilience function may undergo a bifurcation or be- change or human pollution [15] may cause an ecologi- come non-analytic, indicating that the system loses its cal system to cross a tipping point and lead the system resilience by experiencing a sudden transition to a dif- to transfer from one attractor to another undesired at- ferent, often undesirable, fixed point of equation. tractor, from which it would be very hard to go back to However, such one-dimensional approach rules out the original state. its application to many real systems that are usually Due to the diversity and complexity in the system’s multi-dimensional. One solution is to lift (or embed) response to perturbations, the concept of resilience is the nonlinear dynamics into a higher dimensional space 3 Figure 2: Network resilience in one-dimensional systems. A, The system exhibits a single stable fixed point for β > βc (blue) and two (or more) stable fixed points otherwise. B, Resilience function with a first order transition from the desired (blue) state to the undesired (red) state. C, Resilience function with a stable solution for β < βc and no solution above βc. Figure from [12]. where its evolution is approximately linear [33]. Such prerequisites make it possible for us to develop analytic lifting can be achieved by changing the focus from “dy- tools for the resilience of large-scale complex networks. namics of states” to the “dynamics of observables” [34]. Recently, Gao et al. [12] developed a set of analytical A set of scalar observables can measure a system’s state, tools for identifying the natural control and state param- and there are different choices for the observations. If eters of a large-scale complex system, helping derive one could find a set of observables whose dynamics ap- effective one-dimensional dynamic [53] that accurately pear to be governed by a linear evolution law, then the predicts the critical point (tipping point) where the sys- resilience in nonlinear dynamical systems could be de- tem loses its ability to recover. The proposed analyti- termined entirely by the spectrum of the evolution oper- cal framework allows us to systematically separate the ator [35]. The two primary candidates for the study of roles of the system’s dynamics and topology. Although dynamical systems via operators are the Koopman oper- the critical point depends only on the system’s intrin- ator [36] and the Perron-Frobenius operator [37], which sic dynamics, three topological characteristics: density, are dual to each other under appropriate function space. heterogeneity, and symmetry can enhance or diminish a The Perron-Frobenius operator represents a picture of system’s resilience. The dimension-deduction method “dynamics of densities”, in which it is hard to compute [12] has been applied to a class of bipartite mutual- invariant densities, while Koopman operator presents a istic networked systems in ecology, arriving at a two- “dynamics of observables”, which are more suitable to dimensional system that captures the essential mutual- physical experiments. Thus, Koopman operator is more istic interactions in the original large-scale system [54]. widely used in analyzing the system’s dynamics, which Based on the theoretical tools for both low- is a linear operator, generalizing linear mode analysis dimensional and large-scale networks above, and the from linear systems to nonlinear systems [34, 38]. Such advanced data analyzing techniques [55], studies on the a linearizing method may cause the system to simplify resilience in real networks from various fields, ranging by shedding its nonlinear nature, but it requires to com- from the natural to man-made world, have been carried pute the eigenfunctions of the Koopman operator [38], out. The goal of this article is to review the advances on which limits this method usefulness to extremely large the resilience in real networks. To achieve this we dis- systems. cuss a series of topics that are essential to understand- Extracting the resilience function of the complex sys- ing of the resilience of networks, such as alternative sta- tems requires the accurate wiring diagram of the sys- ble states (bistability) [56], regime shifts [57] and early tem and a description of the nonlinear dynamics that warning signals [58]. We illustrate these topics accord- governs the interactions between the components. The ing to their application scenarios in ecology, biology, as emergence of network science provides powerful tools well as in social and infrastructural systems. to characterize the structure of large-scale complex networks [39, 40, 41, 42], such as the Internet [43], power • Ecological network resilience: In ecology, species grids [44, 45], genome-scale gene regulatory networks extinction/co-existence is a critical issue attracting [46] and metabolic networks [47]. The nontrivial topol- lots of attentions. Plenty of studies have been ogy of real-world networks have been uncovered and carried out to predict the thresholds and tipping characterized in the past two decades [48]. In addi- points at which certain species go extinct. These tion, the accumulations of massive data and rapid devel- studies are used to focus on the analytical solutions opment of computational methods make it possible to of low-dimensional systems or the numerical sim- identify and predict the exact forms of dynamic models ulations of high dimensional systems. Recently, directly from empirical data [49, 50, 51, 52]. These two there are more studies on the analytical prediction 4 of resilience in high-dimensional systems. stresses, societal stresses, or acute events [60]. Further- more, the topics related to resilience, such as alternative stable states, also exist in other disciplines. In material • Biological network resilience: In biology, many science, for example, most types of solid matter have living systems exhibit drastic state shifts in a single stable solid state for a particular set of condi- response to small changes in environmental tions. Yet Yang et al. [61] describe a material composed parameters, leading to diseases or apoptosis. We of a polymer impregnated with a super-cooled salt so- will review the recent studies on predicting tipping lution, termed as sal-gel, that assumes two distinct but points and discovering early-warning signals in stable and reversible solid states under the same con- organisms, which could help prevent or delay the ditions for a range of temperatures. Besides the vari- onset of diseases. ous definitions of resilience itself, there are conceptual overlaps between resilience and other concepts, such as robustness and stability, which we will discuss in this • Social network resilience: Behavior of social review to demonstrate that these are distinct concepts. groups of humans and social animals are likely to The advances reviewed here will advance the readers’ exhibit tipping points. We first make distinction understanding of the complex systems surrounding us, between cultural and survival resilience. Then, we and enable the readers to design more resilient infras- summarize the concepts behind tipping points and tructure or social systems. show that they arise in both types of resilience. We also show instances in which they are likely to occur in human and animal societies. Studying 2. Tipping points in ecological networks tipping pints may open up new lines of inquiry in The structure and function of the ecological systems behavioral ecology and generate novel questions, assembled by the interacting species are subjected to in- methods, and approaches to human and animal ternal or external perturbations, such as human-induced behavior. pressures and environmental changes [4, 28, 62, 63, 64, 65]. These ecological systems need to respond to • Infrastructural network resilience: Most infrastruc- such perturbations by adjusting their activities to retain tural network systems are considered as recover- their basic functionality. Unfortunately, due to climate able due to effective human interventions. The tra- change and increasing human activity, even a small per- ditional concept of “engineering resilience” mea- turbation may cause an ecosystem to cross the tipping sures the time it takes for a system to recover or the point and reach an unexpected domain of attraction. relative change in its recovery back to equilibrium This transition between states uncover the alternative after a disturbance. Very recently, a few groups are stable states [66, 28] or phase shifts [67, 68, 69]. The studying the system resilience and prediction of the ability to predict the tipping point of the alternative sta- tipping points in engineering systems. For exam- ble state and the phase shift is crucial for ecosystem ple, in transportation systems, certain traffic con- management. Though both terms show the changes in gestions may be avoided if we could give effective the ecosystem in response to disturbances, they repre- early-warnings. sent different characteristics change in dynamical systems. As shown in Fig.3: 1) alternative stable states Resilience problems are ubiquitous, with wide appli- implies at least two different stable equilibria can occur cations to many other disciplines besides the four do- at the same parameter values (i.e., environment) over mains discussed above, such as environmental science, at least some of the range, where hysteresis [70] of- computer science, management science, economics, poten appears; 2) a phase shift has but a single state (al- litical science, business administration, and phycology beit changing at the threshold) under all parameter val- [25]. For instance, the extensively studied psycholog- ues. Transitions between states are changes from the ical resilience has multiple definitions with adversary dramatic qualitative changes in the former case, while and positive adaptation being two core concepts, which simple quantitative changes in the latter. characterize the ability to mentally or emotionally cope Thus, before predicting the state of such a dynami- with a crisis [59]. For business and economic systems, cal behavior, an important question remains about the resilience is defined as their capacity to survive, adapt landscape of the system whether it “has a single val- and grow in the face of change and uncertainty related ley or multiple ones separated by hills and watersheds” to disturbances, whether they are caused by resource [31]. If the former, the dynamical system has a unique 5 Figure 3: Schematic of different types of state transitions in ecosystem states. (A) The state in an ecosystem response linearly to the changes in conditions; (B) A continuous and gradual “phase shift” from “upper” to “lower” mutually exclusive states as the background environmental parameter increases; (C) A limiting case of sudden regime shift between two mutually exclusive states; (D) A sudden catastrophic shift between multiple stable states in a hysteretic system with different threshold for forward and backward shifts. Source: Figure is modified from [71]. attractor, to which the system will always evolve from 2.1.1. Mathematical models for multiple stable states all initial conditions following any disturbance, show- Plenty of theoretical models [31] have demonstrated ing no correlations to the past states. If the latter, the the behaviors of complex ecological systems with mul- state to which the system settles depends on the initial tiple stable states. From the modeling perspective, alter- conditions; the system may return to the original state native stable states might occur through state variables under small perturbations, but evolve into another at- shifts [56]. Thinking about the shift between alterna- tractor following large perturbations. tive stable states is aided by the ball-in-cup analogy out- lined in Fig.4. All conceivable states of the system can 2.1. Multiple stable states and resilience in ecosystems be represented by a surface, with the actual state of the system (for example, the abundance of all populations) Whether alternative stable states exist in nature has as a ball residing on it. The movement of the ball can been the subject of a historical debate [72, 73, 74, 75]. be anticipated from the nature of the surface: the ball al- The debate regarding what constitutes evidence for always rolls downhill without external intervention. In the ternative stable states arose because there are two differ- most straightforward representation of alternative stable ent contexts in which the term “alternative stable states” states, the surface has two basins, with the ball residing is used in the ecological literature [56]. One excludes in one of them. Valleys or dips on the surface represent the effect of environmental change and regards the en- domains of attraction for a state. The question is, how vironment as fixed in some sense [76, 77]. Another does the ball move from one basin to the other? There focuses on the effect of environmental change on the are two ways: either move the ball (Fig.4, left) or al- state of communities or ecosystems [31]. For exam- ter the landscape upon which it sits (Fig.4, right). The ple, Connell et al. [78] suggest that alternative sta- first of these requires substantial perturbation to the state ble states do not exist in systems untouched by hu- variables (for example, population densities). The latter mans, while Dublin et al. [79] and Sinclair et al. [80] envisions a change to the parameters governing inter- point out that human being is a part of the dynamics actions within the ecosystem, such as birth rates, death of systems. Here we would not tread on the ques- rates, carrying capacity, migration, or per capita predation of whether the human is or is not natural parts of tion, which can be changed by human interventions or ecosystems, but just observe that human does change natural disasters. the states of ecosystems [81]. Much of the literature Simple theoretical analyses predict multiple stable over the last 50 years have shown increasing empirical states for 1) single species dynamics via the Allee ef- support of alternative stable states gathered by ecolo- fect, 2) two-species competitive interactions character- gists [72, 31, 79, 82, 66, 73, 83, 28, 56, 84, 85, 86, 65], ized by unstable coexistence, 3) some predator-prey since 1960s [87]. Transitions among stable states are interactions, and 4) some systems combining preda- used to describ many ecosystems, including semi-arid tion and competition [66]. The theoretical models of rangelands [82], lakes [73], coral reefs [28], and forests multiple stable states are usually represented by a sys- [88]. Moreover, plenty of theoretical models have been tem of dynamical equations characterizing the evolution proposed to account for the alternative stable states in of the species’ states. If the equation describing the ecosystems. Next, we will review the mathematical transformation of the state in the ecosystem is nonlin- models of multiple stable states. ear, there may be multiple stable states with all species 6 present; If the system of equations governing the species increases to a peak value and then decreases due to the is linear, then only one stable state exists with all the competition on the resources. When G(V) decreases to species present. Yet, there may exist other stable states zero, the biomass V reaches its maximum value K. The with some of the species absent [87]. For perturba- per capita consumption function c(V) increases with V tions to state variables, state shifts have most often been when V is low; and saturates to some constant V0 due achieved experimentally by species removal or addition to the limited intake capacity and digestion rate of the [72, 89]. For example, overfishing is a classic case in herbivore. The maximum biomass K of vegetation and which a new interior community state may arise sim- the saturate value V0 of herbivores are crucial factors ply through changes to the size of the fish population for the final equilibrium point. Define α = V0/K whose [56]. Two species Lotka-Volterra competition is a case value determines how many stable states that a grazing where the interior coexistence equilibrium may be un- ecosystem has. stable, and alternative states arise through the extinc- Functions G(V) and C(V) could have different ex- tion of one population [89, 76]. The parameter changes plicit forms. Among them, the best known form of G(V) [28, 84] may alter the location of a single equilibrium is the logistic function G(V) = rV(1 − V/K), where r point, or may transiently result in destabilization of the is a specific growth rate describing how quickly V ap- current state, permitting the system to reach an alterna- proaches the equilibrium, and C(V) is the “Type III” 2 2 2 tive, locally stable equilibrium point, which may or may consumption function C(V) = βHV /(V0 + V )[90]. not exist before the parameter perturbation. These re- Then the overall grazing model is sults support the notion that complex ecosystems maintain multiple stable states. In the following, we will re- dV V βHV2 = rV(1 − ) − . (2.1) dt K 2 2 view four sample models with multiple stable states in V0 + V low-dimensional ecosystems. Introducing the rescaled variables X = V/K, τ = rt and γ = βH/rH, the equation above takes the form

dX γX2 = X(1 − X) − . (2.2) dτ α2 + X2 √ If parameter value α < 1/3 3, the system shows three equilibria: the low and high biomass equilibria are stable, and intermediate biomass equilibrium is unstable. Near the resilience thresholds Hc2 as shown in Fig.5, a small increase in stocking rate may move the system from the high biomass equilibrium to the low biomass equilibrium. A minimal model. In an ecosystem, the dynamical changes of its state usually show a hysteresis transition, such as desertiﬁcation and lake eutrophication. As shown in Fig.6, if the system is on the upper branch, Figure 4: Two-dimensional ball-in-cup diagrams showing (left) the once the control parameter exceeds the threshold T2, it way in which a shift in state variables causes the ball to move, and will collapse into a stable state in the lower branch. If (right) the way a shift in parameters causes the landscape itself to one tries to restore the state in the lower branch into a change, resulting in movement of the ball. state in the upper branch, the control parameter has to Source: Figure is reproduced from [56]. be lower than another threshold T1 that is much smaller Grazing ecosystems. Suppose there is only one popula- than T2. The following minimal model describes the tion of herbivores with a constant density, H, in a graz- change over time of such “unwanted” ecosystem prop- ing ecosystem [31], and they are sustained by vegetation erty x [28]: whose biomass is V. The growth rate of the vegetation dx rxp is G(V) that is a function of V in the absence of graz- = a − bx + , (2.3) dt xp + hp ing. The herbivores consume the vegetation at a net rate C(V) = Hc(V), where c(V) denotes the consumption where x is the state variable, and r is the control pa- rate per capita. As the biomass V increases, G(V) ﬁrst rameter. The parameter a represents an environmental 7 K

V State

H H c1 c2 H Control Parameter

Figure 5: The equilibrium biomass V as a function of the density, H, Figure 6: The hysteresis in an ecosystem. If the stable state is at the of herbivores with α = 0.08. There are two resilience thresholds, Hc1 upper branch and the control parameter slightly exceeds the threshold and Hc2, in the system. For H between the two thresholds, an T2, the system will collapse rapidly to the lower branch, which is unstable state (green dash-dotted line) and two stable states appear, what we usually want to avoid in the real world. If the system is in a and the system may move to the low or high equilibrium, depending lower equilibrium branch, and we want to restore it back into a on whether the initial value of V lies below or above the dash-dotted higher equilibrium, then we need to decrease the control parameter to “breakpoint” curve. the value smaller than the threshold T1. Source: Figure is modified from [31]. Source: Figure is modified from [28]. factor that promotes x, and parameter b is the decay rate onize dead coral by spreading vegetatively over algal of x. For r = 0, the model has one single equilibrium at turfs. Coral naturally dies at a rate of dC, and Algal x = a/b. Otherwise, the last term representing the Hill turfs arise when macroalgae are grazed (gM/(M + T)). function can cause alternative stable states. The expo- Besides, coral recruit to and overgrow algal turfs at the nent p determines the steepness of the switch occurring combined rate r. Figure7 shows the phase plane of tra- around the threshold h, and the higher p is, the stronger jectories of the system from a given initial state to its the hysteresis is as measured by the distance between equilibrium states, which reveals all the possible stable thresholds. and unstable equilibrium of the system. Coral reef model. After experiencing mass disease- A vegetation-algae model. In shallow lakes, algal induced mortality of the herbivorous urchin Diadema growth increases the enrichment of turbidity, while veg- antillanrum in 1983 and two framework-building etation harms turbidity. Such feedbacks between al- species of coral, the health of reefs in the Caribbean gal, vegetation, and turbidity result in multiple stable have been heavily negatively affected, showing a phase change from coral- to algal-dominated state. Numby et al. [91] discover multiple stable states and hystere- 1 1 A B sis by using a three-state analytical model with corals, 0.8 0.8 C C macroalgae and short algal turfs. Let the coverage of 0.6 s 0.6 s corals, algal turfs and macroalgae be denoted as C, T Coral 0.4 Coral 0.4 CMT u and M, respectively. Assuming that the sum T + M + C 0.2 0.2 T M T M u s u s is constant and equal to one, the dynamics of the reef 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 can be described by two equations: Macroalgae Macroalgae dM gM Figure 7: The equilibrium covers and trajectories over time of = aMC − + γMT, dt M + T macroalgae and corals with grazing intensities g of 0.3 (A) and 0.1 (2.4) (B). Equilibrium covers are represented by black circles. Trajectories dC = rTC − dC − aMC, of system states are represented by the lines beginning at different dt initial covers (red circles) and tending towards stable (denoted by subscript ‘s’) rather than unstable (denoted by subscript ‘u’) where T can be expressed as 1 − M − C. The term equilibria. The parameters are set to a = 0.1, r = 1, γ = 0.8 and −aMC means that macroalgae can overgrow corals, and d = 0.44 [92]. γMT captures the phenomenon that macroalgae col- Source: Figure is modified from [91]. 8 states. Scientists employ the multiple stable state mod- exhibit four key attributes [95]: 1) abrupt state shifts els to understand the transition of states in shallow tem- in time series data, 2) sharp spatial boundaries between perate lakes, showing the regime shift between clear- contrasting states or habitat units, 3) multimodal fre- water states that are dominated by vegetation and tur- quency distribution(s) of key variable(s) with each mode bid states dominated by algae [93, 73]. Among the corresponding to an alternative stable ecosystem state, proposed models [93], a vegetation-algae model (also and 4) a hysteretic response to a changing environment called vegetation-turbidity model) [73] captures the in- [71]. Despite the difficulties, the accumulated body teractions between the growth of planktonic algae (A) of empirical evidence with multiple stable states shows and abundance of vegetation (V), illustrating the poten- that such states exist in natural ecosystems [72, 72]. In tial for alternative equilibria in shallow lakes: the following, we will review several other empirical examples of ecosystems [81] where multiple stable states dA N h = rA v − cA2, have been postulated to exist. dt N + hN V + hv p (2.5) 10 10 h A A B V = , shallow p p 8 8 A + hA 6 6

where r is the maximum intrinsic growth rate of algal algae algae 4 4 deep turfs (A), and parameter c is a competition coefficient. The algal growth increases with the nutrient level (N) 2 2 and decreases with the abundance of vegetation (V) in 0 0 -8 -7 -6 -5 -4 -8 -7 -6 -5 -4 simple Monod relations, with hN and hV denoting the nutrients nutrients half-saturation constants. Note that Monod function is Figure 8: (A) An equilibrium algae abundance in shallow lakes as a a particular case of Hill function [94] with the power function of nutrient level, which shows multiple stable states. The exponent being 1. Vegetation abundance in a shallow solid lines represent stable equilibria and the dashed line represents lake declines with algal biomass in a sigmoidal way, unstable equilibria. (B) Equilibrium density of algae in deep lake gradually declines as a function of the nutrient level decreases that with hA denoting a half-saturation constant. The Hill does not show multiple stable states. coefficient of p shapes the relation between vegetation Source: Figure is modified from [73]. abundance and algal biomass. A high value of p causes change shape to approach a step function representing Shallow Lakes. The existence of qualitative differences the disappearance of vegetation from a shallow lake of in the state of lakes has long been recognized [81]. Shal- homogeneous depth around critical algal biomass where low lakes can have two alternative equilibria: a clear turbidity makes the average depth of the lake unsuitable state dominated by aquatic vegetation, and a turbid state for plant growth [73]. characterized by high algal biomass [96, 73, 97]. Many In the shallow lakes, the equilibrium density of al- ecological mechanisms are probably involved, and each gal changes following the nutrient level changes, show- of these states is relatively stable due to interactions ing a catastrophic fold as shown in Fig.8A. The algal among nutrients, the types of vegetation, and light pen- biomass has three equilibria in a certain range of the etration. The observed trends are: 1) rising nutrient nutrient levels, where one is unstable, and two are sta- level increases turbidity, 2) vegetation reduces and also ble. In constrast, in deeper lakes, the vegetation abun- depends on turbidity, and 3) light penetration limit the dance gradually declines as turbidity increases (Fig.8B) growth of vegetation below certain depth. In the clear- [93, 73]. This indicates that the multiple stable states water state that is dominated by macrophytes, vegeta- arising from the interactions captured by this model are tion can stabilize such state in shallow lakes up to rel- limited to shallow lakes. atively high nutrient loadings [96, 73]. Once a system has switched to a turbid state dominated by phytoplank- 2.1.2. Empirical examples of ecosystems with multiple ton, the system stays in such a state unless it receives stable states a substantial nutrient reduction, which enables recolo- There are more theoretical studies on multiple sta- nization by plants. ble states than empirical studies. However, moving Scheffer et al. [73] review evidence for the from theory to practice is not straightforward. While shift in shallow temperate lakes between clear- the meaning of stability and equilibrium points are very water, macrophyte-dominated states, and turbid, clearcut in theory, it is not so in nature [70]. Nature phytoplankton-dominated states. Transitions between ecosystems with alternative stable states are expected to states can be mediated by trophic relationships where 9 fish and nutrients are the primary drivers [81]. On one changes associated with multiple stable states and hys- hand, the transition from a turbid to a clear state can teresis [70]. Mumby et al. [91] use a mechanistic model be accomplished by decreasing stocks of planktivorous of the ecosystem to discover multiple stable states and fish, which decreases the predation on herbivorous verify it with Hughes’ empirical data, and point out that zooplankton. Consequently, populations of herbivo- both theoretical model and empirical data are far more rous zooplankton increase, leading to an increase in consistent with multiple attractors than the competing herbivory and a reduction in phytoplankton biomass. hypothesis of only a single, coral attractor [101]. Besides, increased light penetration and available Multiple stable states in ecological systems occur nutrients result in the establishment of vegetation [96]. when self-reinforcing feedback generate multiple stable On the other hand, shifts from the clear to turbid state equilibria under a given set of conditions [102, 103]. In can result from overgrazing of benthic vegetation by the coral-dominated state, a decline in coral cover liber- fish or waterfowl [96]. Also, the points where these two ates new space for algal colonization. Once maximum shifts happen are not the same, forming a hysteresis levels of grazing have been reached, further increases phase transition, as shown in Fig.9. in the grazable area reduce the mean intensity of grazing and increase the chance that a patch of macroalgae will establish itself, ungrazed, from the algal turf. The resulting rise in macroalgal cover will reduce the availability of coral settlement space and will increase the frequency and intensity of coral-algal interactions. The resulting diminishing of coral recruitment will reduce the growth rate of corals and will cause limited mortality. This in turn will further reduce the intensity of grazing, thereby reinforcing the increase in macroalgae [91], leading to a stable macroalgae-dominated state, as shown in Fig 10.

Figure 9: Shallow lakes are often shown to respond non-linearly to eutrophication (blue dashed) and in many cases hysteresis have been assumed (orange dotted). Hysteresis is a nonlinear behavior where the state of a lake does not only depend on its present input but also on the prior state. As a result, two stable states could occur given the same conditions. Source: Figure from [97, 98].

Coral Reefs in marine ecosystems. Multiple stable states not only exist in the shallow lakes but also appear Figure 10: With a big enough disturbance, a coral dominated state in other aquatic systems, such as coral reefs, soft sedi- goes across a threshold and transit into an algae dominated state. ments, subtidal hard substrate communities, and rocky Source: Figure from [104]. shores in marine ecosystems. Among those coral reefs are the most well-known and best-documented cases Wetlands. Each wetland is exposed to variations in [70]. Coral-dominated and macroalgae-dominated soils, landscape, climate, water regime and chemistry, states have long been discovered on reefs [99, 68, 100]. vegetation, and human disturbance. Multiple stable Nevertheless, whether they can be called alternative sta- states appear due to the interactions between plants, an- ble states has been a highly effective controversial topic. imals, and environmental changes. For example, due For example, Dudgeon et al. [86] propose that the to nonlinear and coupled ecological, hydrological, and data from fossil and modern reefs support the phase- geomorphological feedbacks, multiple stable states are shift hypothesis with single stable states. Moreover, established in different kinds of coastal tidal wetlands: most studies on the transition from coral-dominated marshes, mangroves, deltas, seagrasses [71]. We show to algal-dominated states do not distinguish between the global distribution of these four kinds of wetlands simple quantitative changes and dramatic qualitative in Fig 16. In salt marshes, evidence from field observa- 10 removal of grazing, which is vital for the management of the ecosystem. While on savanna rangelands, Walker [115] and Ludwig et al. [116] identify alternative stable states as either woody/grass coverage or woody thicket. The transition between these states is often triggered by grazing pressures that remove either drought-tolerant or perennial grasses [116, 115]. If grazing pressures are high, the perennial grass abundance is decreased, leading to an increased abundance of woody plants. Once Figure 11: Global distribution of seagrass meadows, salt marshes, the woody community is established, fires burn less fre- mangroves, and major deltas. quently, and the woody community persists for decades. Source: Figure from [71]. As we mentioned earlier, human intervention is a significant cause of multiple stable states in ecosystems. For example, the woodlands of the Serengeti- tions has documented possible occurrences of alterna- Mara ecosystem in East Africa present multiple stable tive stable states: bare sediment state, vegetated state, states as a result of elephant and human-induced fire: and states with distinct marsh vegetation communities without other external perturbations, such as fire, ele- [105, 106, 107, 108]. For example, multimodal fre- phants were unable to cause the vegetation to transition quency distributions, abrupt state shifts, and hysteric re- from woodland to grassland. Once a fire changes, the sponse make it likely that the system will persist in the vegetation from woodland to grassland, and elephants new state rather than return to the prior state. This was can hold it in the grassland state [79]. also empirically observed in a case of pronounced elevation distribution changes in areas that shifted from bare 2.1.3. Ecological resilience and engineering resilience flats to vegetated marshes in 1931−1992 in the West- Related to multiple stable states (or alternative stable ern Scheldt estuary [109]. In freshwater marshes and states), a crucial feature called resilience has been at- deltas, feedbacks between sedimentation and vegetation tracting considerable attention. Resilience defines the might form alternative stable ecosystem states. For ex- ability of a system to retain its basic functionality when ample, after flooding disturbance, only plants with roots internal change or external perturbations occur. In 1973, longer than a threshold could survive, so the persisting Holling [4] introduced the notion “resilience” to charac- root length is related to sedimentation [110]. In man- terize the degree to which a system can endure perturba- grove systems, one mechanism of forming multiple sta- tions without collapse or shifting into some new stable ble states has been identified in the field: runaway sed- states. Since then, different authors used “resilience” imentation. Sediment may be preferentially deposited in different ways, leading to a great deal of confusion in large amounts. For example, the presence of man- about this term. As shown from Fig4, Beisner et al. grove biomass could promote sudden depositions dur- [56] associated resilience with the features of the basin ing cyclones, hurricanes, or tsunamis [111, 112], and lo- that act to retain the states of species (the location of the cal sedimentation. In seagrass meadows, the vegetated ball). These features are the steepness of the slope and state could persist due to positive feedbacks: vegetation the width of the basin. From one point of view, steep- could decrease water turbidity, enabling the submerged ness of the sides of the basin affects the return time of seagrass plants to intercept enough light to perform pho- the ball to the lowest point in the basin, and the rate, at tosynthesis, which stabilizes the bottom and reduce sed- which a system returns to a single steady or cyclic state iment resuspension [113]. In contrast, when the water following a perturbation. This point of view is taken by depth increases, the vegetated stable state of a seagrass “engineering resilience” defined by Holling et al. [117]. meadow can quickly and abruptly shift to a bare sedi- Engineering resilience focuses on the behavior of a sys- ments state [114]. tem when it remains within the stable domain that con- Rangelands and woodlands. On North American tains this steady-state [117], which means that it con- rangelands, lower successional stable states occur in centrates on stability near an equilibrium steady state. sagebrush and other shrub-dominated vegetation types From the other point of view, the width of the basin in the Great Basin. Laycock [82] pointed out that could also affect the movement of the ball, and the ball counter to the single stable state (climax) assumption, if can only move out of a basin if it undergoes sufficiently a vegetation type is in a stable lower successional state, large perturbations. The size of the perturbation to state it usually does not respond to change in grazing or even variables affect the likelihood of the ball to escape from 11 a basin. This point of view was assumed by “ecolog- changing its structure, identity, and functions [23]. It ical resilience” [21]. Ecological resilience emphasizes focuses on the magnitude of the disturbance that a sys- conditions far from any equilibrium steady state, where tem can withstand without shifting to another regimes instabilities can flip a system into another regime of be- [118]. Hence, the corresponding system simultaneously havior. monitors and reorganizes the processes that govern the Engineering resilience is probably the most fre- system’s behavior to accommodate or resist the distur- quently invoked meaning or definition of resilience bance [23]. Note that this term implies the potential ex- [118, 119]. It assumes that the system is in equilibrium istence of multiple stable states. If the perturbation is before disturbances are introduced and the resilience small, the system could go back to the pre-perturbation is defined in terms of the stability of a system near state. However, if a shock perpetuates changes in con- its steady-state. Hence the resilience denotes the sys- ditions that exceed some intrinsic threshold, the system’s ability to recover back its pre-perturbed steady- tem changes regimes such that the structure or func- state [120]. The concept of engineering resilience mea- tion of the system is fundamentally different. More- sures the time it takes for a system to recover or the over, when a threshold is crossed, the return is difficult relative change in its recovery back to equilibrium af- [133, 134, 135]. For example, many ecological systems ter a disturbance. For example, the resilience triangle may undergo abrupt transitions to alternative regimes paradigm based on lost functionality and recovery time [28, 136]. Such transitions are often irreversible and has been used to quantify system’s resilience [121, 122]. widespread in human health [137], the economy [138] Rose [123] used the time-dependent aspects of recovery and the environment [31]. In the psychological liter- in the definition of dynamic resilience. Ouyang et al. ature, ecological resilience has been recognized as an [124] assessed the resilience of interdependent infras- individual’s capacity to be robust, demonstrating confi- tructure systems during a period consisting of a damage dence in one’s strengths and abilities, and being stoical, propagation stage and recovery stage. In the psycho- resourceful, and determined as one navigates through logical literature, engineering resilience has been used key challenges across and within one’s life [139, 140]. to characterize an individual’s capacity to rebound or In ecology and biology, ecological resilience illustrates ’bounce back’ to their original state following stressful the resistant bacteria responses to water column distur- experiences [125, 126, 127, 128]. In natural systems, re- bances in microbiology [129], the strong responses of silience has been used in the explanation of equilibrium receptors and mechanisms in biology [141], the perma- states in water column disturbances in microbiology, nence of farm systems in ecology [142]. The concept of dynamic restoration of coastal dunes, and the restora- ecological resilience implies the need for predicting the tion of critical ecosystem services [129, 130, 131]. critical points ( or ”tipping points”), at which the system loses its resilience, and to uncover the mechanisms underlying such critical transitions.

Figure 12: Schematic diagram of a resilience triangle. The system functionality degrades after the crisis, but it recovers gradually to return to its level before the crisis in the long run. The recovery will be rapid for a system with a high resilience but slow in case of a low resilience. A resilience triangle is the area of degradation in quality of recovery. Source: Figure from [132]. Figure 13: Schematic diagram of ecological resilience and transitions Ecological resilience, or system resilience, quanti- between multiple stable states. fies a system’s ability to absorb a disturbance without Source: Figure from [143]. 12 In short, engineering resilience focuses on effi- relationship between the states of an ecosystem and con- ciency, constancy, and predictability, while ecological ditions is nonlinear: a continuous and gradual “phase resilience focuses on persistence, change, and unpre- shift” from “upper” to “lower” mutually exclusive states dictability [21, 81]. They are alternative paradigms as the background environmental parameter increases characterizing two different fundamental aspects of a (Fig.3B) [71]. Both in the linear and smooth nonlin- system’s ability to maintain its functionality. Engineer- ear cases, there is only one single equilibrium state, and ing resilience is most useful for a system that exists near the ecosystem could return to the former state by revers- a single or global equilibrium condition. In the tradi- ing the control parameter to the previous level [101]. tions of engineering, systems are designed with a single Some ecosystems may be quite inert over a specific operating objective, which is usually associated with a range of conditions but respond abruptly when the con- single steady-state [144]. It means that we can always trol parameter approach a threshold, showing a sudden repair the system that can return to its regular operation shift between two mutually exclusive state (Fig.3C). even under large perturbations. Resilience in the fields These three types of state transitions are called “phase such as engineering, physics, control system design, or shift” or “regime shift”, which is characterized by dom- material engineering means engineering resilience [81]. inant populations of an ecological community respond- For a system that contains multiple stable states, ecolog- ing smoothly and continuously along an environmental ical resilience is a more meaningful measure for ecolog- gradient until a threshold is reached, shifting the com- ical systems. In ecology, resilience has been defined as munity to a new dominant or suite of dominant popu- “a measure of the persistence of systems and of their lations. In any given environment, there is at most one ability to absorb change and disturbance and still main- stable state [86], except in the threshold point where the tain the same relationships between populations or state system shows an abrupt shift and may have more than variables” [145]. Note that there are some other studies one attractor in principle. In some cases, the abrupt about engineer resilience in ecological systems. Once phase shift may be largely irreversible. For example, the disturbances exceed a tipping point, the system will some cloud forests were established under a wetter rain- shift into another regime. Next, we will review works on fall regime thousands of years previously, and neces- both empirical and theoretical analysis and predictions sary moisture is supplied through condensation of water on phase shifts in ecosystems. from clouds intercepted by the canopy. If the trees are cut, this water input stops, and the resulting conditions 2.2. State transitions in ecosystems can be too dry for recovery of the forest [88, 64]. A crucial different situation arises when the ecosys- Many ecosystems are exposed to internal and external tem response curve is ‘folded’ backward, forming mul- perturbations. The perturbations can be gradual changes tiple stable states for a specific range of parameters in climate, nutrients or toxic chemicals loading, habitat [28, 148], with the parameter describing an external fragmentation or biotic exploitation, and even precipi- condition rather than an interactive part of the system, tating events, such as hurricane and earthquakes [28]. or the change in such condition being very slow relative The responses of ecosystems to perturbations are non- to the rates of change in the system. Vasilakopoulos et trivial: 1) for the same perturbation, different ecosys- al. [148] provide empirical evidence for the occurrence tems show different types of responses, which mainly of a fold bifurcation in an exploited fish population, as includes “phase shift” and “multiple stable states”; 2) shown in Fig. 15. In such a case, the ecosystem has for the same ecosystem, it responds differently to differ- two alternative stable states, separated by an unstable ent perturbations and may even differ its response to the equilibrium that marks the border between the ‘basin of repetition of the same perturbation due to the existence attraction’ of the states, showing as the green dashed of multiple stable states. line in Fig.3D. When conditions or the control parameter changes 2.2.1. Differentiation between phased shift and multiple gradually, the abrupt shift of states occurs at the criti- stable states cal point. In such a case, the “upper” and “lower” states When external conditions change gradually, even lin- need not be mutually exclusive, and they could coexist early, with time [146], the state of some highly resilient in a specific range of conditions. The two critical points ecosystems may respond in a linear way to such trends, – (1) the ecosystem shifts from “upper” to “lower” as shown in Fig.3A. For example, in the coral reef sys- states, and (2) the ecosystem shifts from “lower” to tem, there is a simple linear relationship between coral “upper” states – are different. If the ecosystem is in a and fishing [147, 101]. A more common and complex “lower” state, it is difficult to return to the “upper” state. 13 However, the shift can also happen before the critical point, depending on the strength of disturbance and the size of the basin, which together define the resilience of a system [4]. If the disturbance is large enough, the state of an ecosystem may go over the border of the attraction basins and enter another basin even when it did not yet reach the critical point. When the basin is tiny, that is, the system resilience is low, small perturbation could be enough to displace the ball far enough to push it over the hill, resulting in a shift to the alternative stable state. As shown in Fig. 14, since the size of the basin in which the system’s state lies are low near the critical point, the abrupt shift in states may happen even before the critical point F2 under a large enough perturbation. Note that, although we use the terms ‘stable states’ and ‘equilibria’, ecosystems are never stable in the sense that they do not change, and there are always slow trends and fluctuations. Therefore, Scheffer et al. [95] suggest that the words ‘regimes’ and ‘attractors’ are more appropriate to show the dynamics. Since the terms “multiple stable states” and “alternative stable states” have been extensively used in literature, we also will use these terms while keeping in mind that they refer to a kind of relative dynamic balance rather than stable state excluding dynamics. Figure 15: Folded stability landscape and resilience assessment for Barents Sea cod (1949–2009). On the empirical folded stability landscape for Barents Sea cod (A), continuous black lines mark the linear attractors, dotted black line shows the possible extension of the lower branch, dashed grey lines indicate the approximate position of the basin’s borders, and F1 and F2 indicate the tipping points. Colours represent the relative resilience contour interpolated from the relative resilience of each year. Circles and arrows indicate the 1981 population shift. (B) indicates the relative resilience of each year; black and grey lines refer to the old and new states, respectively. Source: Figure from [148].

Both phase shift and multiple stable states are underlying mechanisms to explain the catastrophic shifts in complex systems. Abrupt shifts with alternative stable states are usually related to bifurcation, as Fig.3(d) shows, while phase shifts are not. This makes it is difficult to collect empirical data showing bifurcations under the same conditions. Thus it is not straightforward to differentiate phase shifts from multiple stable states. Fortunately, despite the requirement of extensive time series containing many shifts [149], there are several ap- Figure 14: External conditions affect the resilience of multi-stable ecosystems to perturbation. The stability landscapes depict the proaches to infer whether or not alternative attractors are equilibria and their basins of attraction at five different conditions. involved in a shift [95]. 1) Based on the principle that Stable equilibria correspond to valleys; the unstable middle section all attractor shifts imply a phase in which the system is of the folded equilibrium curve corresponds to a hill. If the size of speeding up as it is diverging from a repeller, a statisti- the attraction basin is small, resilience is small and even a moderate perturbation may bring the system into the alternative basin of cal approach [150] could be used; 2) Another approach attraction. compares the fit of contrasting models with and with- Source: Figure from [28]. out attractor shifts [151, 152], or computes the proba- 14 bility distribution of a bifurcation parameter [151]. Es- silience in ecosystem is global warming [160], which is pecially, for some colonization events, such as in ma- a result of the exhaust of excessive carbon dioxide.The rine fouling communities [72], which once established, increasing temperatures are modifying key physiologi- can be very persistent and hard to replace until the co- cal, demographic and community-scale process [161], hort dies of old age. Unless the new state can persist decreasing the internal resilience in ecosystems. The through more generations by strengthening itself [153], erosion of internal resilience render ecosystems vulner- it seems inappropriate to relate to such shifts as alterna- able to external disturbances [158], and triggered by tive stable states [95]. Note that although “regime shift” them shifts from one state to another undesired state. and “alternative stable states” are different as discussed Loss of kelp forests. A marine heat wave caused the above, they both can be underlying mechanisms behind loss of kelp forests across 2300 km2 of Australia’s abrupt phase transitions. Since for some of those, there Great Southern Reef, forcing a phase shift to seaweed is no apparent evidence to show whether they are re- turfs [162], as shown in Fig. 16. Before December lated to alternative stable states or regime shift, various 2010, kelp forests covered over 70% of shallow rocky literature uses them without marking such distinction reef in the midwest coast of Australia [163]. However, [154, 155, 156]. In the following review of critical tran- only two years later, by early 2013, extensive surveys sitions, we would not distinguish between them either. found that a 43% (963 km2) loss of kelp forests on the west coast. The previously dense kelp forests had dis- 2.2.2. Individual abrupt phase transitions in real appeared, and a dramatic increase in the cover of turf- ecosystems forming seaweeds had been found (Fig. 16). Wernberg The shift from one state to another may result from et al. [162] deduced that such phase shift is caused by either a ‘threshold’ or ‘sledgehammer’ effect [157]. A a extreme heat wave, with the temperatures exceeded state shift caused by a sledgehammer effect, such as the a physiological tipping point for kelp forests beginning clearing of a forest using a bulldozer, usually comes in 2011, and the new kelp-free state was supported by as a result of our expectation. By contrast, the criti- reinforcing feedback mechanisms. In contrast similar cal threshold is reached as incremental changes accu- ecosystem changes have not been observed in the south- mulate and the threshold value generally is not known west coast, since that the heat wave temperatures there in advance, so that state shifts resulting from threshold remained within the thermal tolerance of kelps [164]. effects is usually not anticipated. In other words, un- The previous short-term climate changing events, such desired shifts between ecosystem states are caused by as the large-scale destruction of kelp forests during the the combination of the magnitudes of external forces EI Nino˜ Southern Oscillation event of 1982/83 [165], and the internal resilience of the system. A sudden dra- have mostly recovered as environmental conditions re- matic change is not necessary to be caused by a sudden turned back to normal [166]. However there is no signs sizeable external disturbance. When the system is close of kelp forest recovery on the heavily affected reefs in to the tipping point, even a tiny incremental change in west Australia [161]. What is worse, the current ve- conditions can trigger a tremendous shift, such as the locity of ocean warming is pushing kelp forests toward legendary straw that breaks the camel’s back and tip- the southern edge of the Australian continent [167]. The ping over of an overloaded boat when too many peo- high risk of local extinction of kelp forests would devas- ple move to one side [95]. Abrupt phase transitions in tate lucrative fishing and tourism industries worth more ecosystems are increasingly common as a consequence $10 billion Australian dollars per year [168], and endan- of human activities that erode internal resilience, for ex- ger thousands of endemic species supported by the kelp ample, through resource exploitation, pollution, land- forests of Australia’s Great Southern Reef. use change, possible climatic impact and altered distur- Self-organised patchiness in the arid ecosystem. In an bance regimes [64, 68, 158]. Next we will show three ecosystem, different attracting states can not only exist typical examples of catastrophic shifts in ecosystems. across different time scales [169, 28, 95], but also coex- Fire in Australia. In Australia, overhunting and use ist at the same time across different spatial scales, which of fire by humans some 30,000 to 40,000 years ago re- is usually related to self-organised patchiness and the removed large marsupial herbivores, which resulted in an source concentration mechanisms involved [170]. The ecosystem of fire and fire-dominated plants expanded most prominent example is the arid ecosystem [171], and irreversibly switched the ecosystem from a more where the self-organised patchiness differs in scale and productive state, dependent on rapid nutrient cycling, shape, such as gaps, labyrinths, rings and spots, as to a less productive state, with slower nutrient cycling shown in Fig. 17. This patchiness is a result of posi- [159, 64]. Another cause leading to the erosion of re- tive feedback between plant growth and availability of 15 Figure 16: Regime shift from kelp forests to seaweed turfs after the 2011 marine heat wave. Kelp forests were dense in Kalbarri until 2011 (A),when they disappeared from 100 km of coastline and were replaced by seaweed turfs (B). (C) The habitat transition (lines) coincided with exceptionally warm summers in 2011, 2012, and 2013 (red bars), punctuating gradually increasing mean ocean temperatures over the past decades. Source: Figure from [162]. water. A higher vegetation density allows for higher water infiltration into the soil and lower soil evapora- tion, which could stabilize the vegetation-state. Once the vegetation disappears, bare soil is too hostile for re- colonisation [169, 172]. Thus the present state of the vegetation is not only determined by soil-water distributions, but also depends on biomass–water feedbacks that is determined by its history states [170, 173]. More importantly, the vegetation states on some spatial locations may shift abruptly into a bare state, if rainfall decreases beyond a threshold, which contributes to a more homogeneous bare state in the arid ecosystem. How- ever, on the other side, increased rainfall may not re- Figure 17: Different pattern morphologies in the Triodia spinifex cover the spotted vegetation state, since the concentra- grassland. Gaps (A), labyrinths (B), spots (C), and rings (D). tion of soil water under vegetated patches already disap- Source: Figure from [173]. peared. Similar patchiness across different spatial locations and the catastrophic shifts have been observed in nutrient-poor Savanna ecosystems [174, 175, 176] and state shifts. Peatland ecosystems [177]. In addition to the decreasing rainfall, overgrazing by cattle can also cause catas- 2.2.3. Regime shifts in a connected world trophic shifts from a spotted vegetation state into a de- Research on regime shifts is often confined to dis- certified ecosystem state [178, 116]. Adequate grazing tinct branches of science, reflecting empirical, theoret- management and patchy crop production to conserve re- ical [180], or predictive approaches [29, 181], which sources in marginally arable lands may help to optimize requires an in-depth knowledge of the causal structure productivity, thereby preventing such catastrophic shifts of the system or high quality of spatiotemporal data. [170]. Hence, the research has generally focused on the analysis of individual types of regime shifts, as discussed Abrupt shifts can cause substantial losses of ecologi- above. As humans increase their pressure on the planet, cal and economic resources and require drastic and ex- regime shifts are likely to occur more often, and more pensive intervention to restoring a desired state [179, severely, and more broadly [182, 183, 184, 185]. An 28]. Studies on the abrupt shifts and tipping points important question to ask is whether regime shifts inter- in ecosystems could help understand the failure mech- act with one another and if the occurrence of one will anisms, which is crucial for ecosystem management. increase the likelihood of another or correlate at distant Thus, we should not only focus on the prevention of per- places. All systems on the planet are closely intertwined turbations but also pay attention to sustaining the system across three dimensions: organizational levels, space, in a large stability domain or push the system far away and time. For example, the excessive use of fertiliz- from the tipping points, reducing the risk of unwanted ers, as an inadvertent result of growing more vegetables 16 on land, could eutrophicate the downstream coastal wa- et al. [185] give a more comprehensive analysis of cas- ters that compromises food production from the ocean cading regime shifts within and across scales. [186], with eutrophication being one of three essential In the regime shift database, each entry provides a causes (eutrophication, warming, and consumption) for literature-based synthesis of the key drivers and feed- rapid declines in marine biodiversity [187]. A variety back underlying the regime shift, as well as impacts of causal pathways connecting regime shifts have been on ecosystem services and human well-being, and pos- identified, showing that the occurrence of a regime shift sible management options. For each regime shift, the may affect the occurrence of another regime shift. For database encodes the drivers and underlying feedbacks example, eutrophication is often reported as a regime into a causal loop diagram [192], which is a signed di- shift preceding hypoxia or dead zones in coastal areas rected graph consisting of variables connected by ar- [188], and hypoxic events have been reported to affect rows denoting causal influence [182]. Rocha et al. [185] the resilience of coral reefs to warming and other stres- use 30 regime shifts, where complete synthesis exists. sors in the tropics [189]. Thus, there are potential in- The causal loop diagrams have been curated to con- teractions between regime shifts across systems, which struct coupled regime shift networks shown in Fig 19. should not be studied in isolation under the assumption The authors merge pairs of regime shift networks using that they are independent systems. the following three types of connections between them: Thanks to the accumulation of the empirical data of (i) driver sharing, which is the most common type. regime shifts, an open online repository of regime shift The regime shifts connected that way are correlated in syntheses and case studies have been built, which is time and space, yet they do not have to be independent called the regime shift database [190]. This database [182, 193]; (ii) domino effects, which occur when the currently contains 35 types of regime shifts, more than feedback processes of one regime shift affect the drivers 300 specific case studies based on a literature review of of another regime shift, creating a one-way dependency over 1000 scientific papers, and a set of 75 categorical [181, 193, 194]; and (iii) hidden feedbacks, which only variables about the i) main drivers of change, ii) impacts show when two regime shift networks are combined to on ecosystem services, ecosystem processes and human generate new feedbacks that cannot be identified in the well-being, iii) land use, ecosystem type and spatial- separated regime shifts [186]. temporal scale at which each regime shift typically oc- By analyzing the regime shift networks coupled curs, iv) possible managerial options and v) assessment through these three interaction types, Rocha et al. [185] of the reversibility of the regime shift, the level of un- find that half of the regime shifts may be causally linked certainty related to the existence of the regime shift, and at different scales [195]. The regime shift of networks its underlying mechanism [182]. These regime shift at- coupled via driver sharing describes the co-occurrence tributes could be used to fit statistical models to explore patterns of 77 drivers across the 30 regime shifts ana- the role of cross-scale interactions [185]. For example, lyzed. The drives that the most frequently co-occurred drivers include natural and human-induced changes that in this case were related to food production, climate have been identified as directly or indirectly producing change, and urbanization. Regime shifts are more likely a regime shift [191]. The same driver can induce differ- to share drivers when they use land similarly. The driver ent regime shifts. Note that drivers, dynamics operate sharing is more likely in dynamics that evolve faster outside the feedback mechanisms of the system, thus in time when there is a march of spatial scales. The they are variables independent of the dynamics of the domino effects are not common; evidence of cross-scale system. Direct drivers are those that influence the inter- interactions for domino effects was only found in time nal processes or create feedbacks underlying a regime but not in space. Hidden feedback is more likely to oc- shift. Indirect drivers alter one or more direct drivers. cur in the range of decades to centuries and at the na- By mining the interactions between regime shifts and tional scale. The regime shift networks constructed in drivers, a bipartite network is constructed in Ref [182], Ref. [185] enable researchers to systematically identify and two networks can be projected from it. The first potential cascading effects. is a network of drivers connected by the regime they The cascading effects among regime shifts have also caused. The second is a network of regime shifts con- been modeled as a network of tipping elements in Ref. nected by the drivers they share. These two projected [196], where each node is one tipping element repre- networks are shown in Fig. 18. The analysis of regime- sented by a time-dependent quantity x(t) that evolves driver networks demonstrates that reducing the risk of according to the autonomous ordinary equation regime shifts requires integrated action on multiple di- dx = −a(x − x )3 + b(x − x ) + r, (2.6) mensions of global change across scales. Thus, Rocha dt 0 0 17 Figure 18: Regime shifts-Drivers Network. In the centre is the bipartite network of drivers (left) and regime shifts (right). On the right is the one-mode projection of regime shifts. The width of the links is scaled by the number of drivers shared, while node size corresponds to the number of drivers per regime shift. On the left is the one-mode projection of drivers, with link width scaled by the number of regime shifts for which causality is shared, and node size proportional to the number of regime shifts per driver. Below each projection is the structural statistics analysis. Source: Figure from [182]. where r is the control parameter, while coefficients terglacial condition [204, 205]. This transition was a a, b > 0 and x0 control the position of the system on rapid warm-cold-warm fluctuation in climate between the x-axis. The interactions between tipping elements 14,300 and 11,000 years ago [205]. The significant bi- are modeled as a directed network of a linearly coupled otic changes included the extinction of about half of system of ordinary differential equations the species of large-bodied mammals, several species of large birds and reptiles, and a few species of small dx XN i = −a(x − x )3 +b(x − x )+r +d a x , (2.7) animals [206]. A vital decrease in local and regional dt i 0 i 0 i i j j j=1, j,i biodiversity occurred as geographic ranges shifted in- dividualistically, which also resulted in novel species where d is the coupling strength and ai j are elements in assemblages [207]. Another significant change was a the adjacency matrix. Such a network captures the cas- global increase in human biomass and spread of mod- cading effects between tipping elements and has been ern humans to all continents [208]. The collected ev- applied to the Amazon rainforest. Although cascading idence of these planetary-scale critical transitions sug- effects usually mean that the occurrence of one regime gests that global-scale state shifts are not the cumula- shift could trigger the occurrence of another one, not tive result of many smaller-scale events that originate in all cascading effects reported in the literature are ex- local systems. Instead, they require global-level forc- pected to amplify each other. For example, it has been ing that emerge on the planetary scale and then perco- reported that climate-tipping points can regulate each late downwards to cause changes in local systems [157]. other and reduce the probability of regime shifts in Now the global-scale forcing mechanisms are induced forests [197, 198]. The study on interactions between by the human population growth with attendant resource regime shifts could help developing methods for early consumption [201], energy production, and consump- warning signals detections to predict regime shifts. tion [209], and climate change [204], which cumula- Most of these studies on regime shifts focus on lo- tively impose much stronger forcing than those active cal ecological systems evolving over short time spans at the last global-scale state shift. Thus, the plausibility [29, 199, 200]. However, there also planetary-scale crit- of a future planetary state shift seems high, as shown ical transitions that operate over centuries or millen- in Fig 20, which highlights the need to predict critical nia [201, 157], such as the ‘Big Five’ mass extinctions transitions by detecting early warning signs. [202], the Cambrian explosion [203], and the most recent transition from the last glacial into the present in- 18 Figure 20: Quantifying land use as one method of anticipating a planetary state shift. The trajectory of the green line represents a fold bifurcation with hysteresis. At each time point, light green represents the fraction of Earth’s land that probably has dynamics within the limits characteristic of the past 11,000 year. Dark green indicates the fraction of terrestrial ecosystems that have unarguably undergone drastic state changes. A planetary state shift may occur assuming conservative population growth and that resource use does not become any more efficient. Source: Figure from [157]. Figure 19: Cascading effects across scales. (A) Summary of the statistical results. (B to J) Circular plots showing the mixing matrices of cascading effects [driver sharing, (B), (E), and (H); domino 2.3.1. Data-driven approach for low dimensional sys- effects, (C), (F), and (I); and hidden feedbacks, (D), (G), and (J)] according to ecosystem type, and spatial or temporal scales. tems Source: Figure from [185]. Predicting the tipping points where critical transition occur is extremely difficult because of the state of the system may show little change before reaching the tip- 2.3. Predicting the tipping points in ecosystems ping point [29]. Fortunately, specific generic symptoms Tipping points are critical thresholds in system pa- may occur in a broad class of systems as they approach rameters or state variables at which a tiny perturbation a critical point, which can be used for early warning be- can lead to a qualitative change of the system [196]. fore the catastrophic shift occur. These symptoms arise Once a tipping point has been crossed, and a critical regardless of differences in the details of each system transition occurs, it is extremely difficult or even im- [211]. Similar signals could indicate disparate phenom- possible for the system to return to its previous state ena such as the collapse of an over-harvested population [157]. Thus prediction and prevention of unwanted crit- at ancient climatic transitions. ical transitions in ecosystems is a crucial challenge in Critical slowing down. The most important clue that ecology. Critical transitions, or regime shifts, can result has been suggested as indicator of system getting close from ‘fold bifurcations’ that shows hysteresis [29] or to a critical threshold is related to a phenomenon known more complex bifurcations [210]. For the regime shifts in dynamical systems theory as ‘critical slowing down’ in the latter case, there will be no typical early warn- [212]. It occurs for a range of bifurcations [29]. At fold ing signals. In contrast, the regime shifts due to fold bifurcation points, the dominant eigenvalue characteriz- bifurcations trigger the general preceding phenomena ing the rates of change around the equilibrium becomes that can be characterized mathematically. Those include zero. This implies that as the system approaches such a deceleration in recovery from perturbations (‘critical critical points, it becomes increasingly slow in recover- slowing down’), an increase in variance in the pattern ing from small perturbations [213]. Thus, the recovery of within state fluctuations [29], an increase in auto- rate after a small experimental perturbation can be used correlation between fluctuations [199], and an increase as an indicator of how close a system is to a bifurcation in asymmetry of fluctuations and rapid back-and-forth point, and such perturbation is so small that it introduces shifts (‘flickering’) between states [204]. no risk of driving the system over the threshold. 19 A straightforward implication of critical slowing nomenon of critical slowing down leads to three pos- down by systematically testing recovery rates is suitable sible early-warning signals in the dynamics of a system for theoretical models [214], but impractical or impos- approaching a bifurcation: slower recovery from pertur- sible to use for monitoring the most of natural systems. bations, increased autocorrelation, and increased vari- However, almost all real systems are persistently sub- ance [29]. ject to natural perturbations, and certain characteristic Skewness and flickering. In the vicinity of a catas- changes in the pattern of fluctuations are expected to trophic bifurcation point, the asymmetry of fluctuations occur as a bifurcation is approached in such a system. may increase [224]. Take the fold bifurcation as an ex- For example, since that slowing down causes the intrin- ample, and an unstable equilibrium marks the border of sic rates of change in the system to decrease, the state the basin of attraction as the system approaches the at- of the system at any given moment becomes more and tractor from one side, as shown in the dashed line in more like its past state, thus an increase in autocorre- Fig.6. As the system approaches the bifurcation, the lation in the resulting pattern of fluctuations appear as slope in the basin of attraction (Fig.4) becomes less the system approach the tipping point [215]. Although steep. Consequently, the system will tend to stay in an there are different measurements can be used to quantify unstable point, but not the opposite side of the stable such increase [216, 217], the simplest way is to look at equilibrium [29], leading to an increase in the skewness lag-1 autocorrelation [218, 219], which can be directly of the distribution of states. Besides, the system’s state interpreted as the slowness of recovery in such natural may be moved back and forth between the basins of perturbation regimes [213, 215]. It has been found that attraction of two alternative attractors under stochastic there is a marked increase in autocorrelation that builds forcing, as the border becomes relatively lower when a up long before the critical transition occurs (Fig. 21) critical transition is approaching. Such a phenomenon both in simple models [219] and complex realistic sys- is called flickering [223, 225], which is another early- tems [220]. warning signal, and the system may shift permanently to the alternative state if the underlying slow change in conditions persists. Indicators in cyclic and chaotic systems. The early- warning signs reviewed above exist in systems with underlying attractors corresponding to stable points, but not applicable to cyclic and chaotic systems, where critical transitions are associated with a different class of bifurcation [226], and the early-warning signs are different in this case. For example, the Hopf bifurcation, which marks the transition from a stable system to an oscillatory system [227], is signaled by critical slowing down [228]: close to the bifurcation, perturbations lead to long transient oscillations before the system settles to the stable state. Another class of bifurcations is caused by intrinsic Figure 21: Early warning signals for a critical transition in a time series generated by a model of a harvested population [221] driven oscillations, which bring the system to the border of the slowly across a bifurcation. (A), Biomass time series. (B) shows that basin of attraction of an alternative attractor, and such the catastrophic transition is preceded by an increase both in the bifurcation is called basin-boundary collisions [229]. amplitude of fluctuation, expressed as standard deviation (s.d.) (C), These are usually not associated with particular proper- and in slowness, estimated as the lag-1 autoregression (AR(1)) coefficient (D), as predicted from theory. ties related to stable or unstable points that can be ana- Source: Figure from [29]. lytically defined. Yet, the dynamics may be expected to change characteristically before collisions occur, such Another possible consequence of critical slowing as increased autocorrelation between states, so that os- down in the vicinity of a critical transition is the in- cillations may become ‘stretched’ [230]. Also, there is creased variance in the pattern of fluctuations [222, 29]. the phenomenon of phase locking between coupled os- Because the critical slowing down could reduce the abil- cillators, and alternative attractors are often involved, ity of the system to track the fluctuations [223], it would when the corresponding bifurcations are associated with increase the standard deviation of the stationary dis- critical slowing down [231]. For example, rising vari- tribution versus input fluctuations. In short, the phe- ance and flickering occur before an epileptic seizure, 20 showing a phenomenon associated with the phase lock- 2.3.2. Model-driven approach for systems with net- ing of firing in neural cells. worked dynamics

Spatial patterns as early-warning signals. Early- There are plenty of empirical examples of regime warning signals arise not only in time series but also shifts with tipping points in ecosystems, and a lot of in some particular spatial patterns in the vicinity of a theoretical models had been proposed to ascribe the critical transition. The spatial term here is not limited underlying mechanisms. Yet, it remains a challenge to physical distances between two elements, such as the to predict the tipping points and regime shifts, due to habitat patches in a fragment landscape [232, 233], but the great complexity of ecological systems, which usu- also could be used to represent the functional associa- ally is reflected in the high dimensionality (large num- tions or interactions between two entities, such as the ber of species and interactions), stochastic and non- connections between two people in the social network linear nature dynamics, and uncertainty of initial con- or two functionally related financial markets [234, 235]. ditions or drivers [238]. Despite this complexity, it In many systems, each entity tends to take a similar state is possible to predict regime shifts through resilience of the connected entities. Moreover, the phase transi- indicators, e.g., the statistical measures of some key tions in such systems may occur like that in ferromag- ecosystem variables [239]. For instance, ecologists netic materials, where individual particles affect each have used a combination of models and observations others’ spin [211]. When such systems approach the from long-term datasets or short-term experiments to tipping point, the distribution of the states of the entities identify early-warning signals before the critical tran- may change in particular ways, such as a general ten- sition occurs [240, 210, 241]. In these models, initial dency towards increased spatial coherence [211], which conditions are informed by the first principle and emis measured by cross-correlation among entities. pirical data, the drivers are incrementally or dramatically altered, and the ensuing changes to the system are Although many systems have similar early-warning recorded. For ecosystems with certain types of dynam- signals, no general spatial patterns are fitting for all ics, this approach could successfully detect the early- systems. It is essential to know which class of sys- warning signs. However, some other ecosystems, es- tem is involved as we interpret spatial patterns. For pecially those having multiple attractors or the poten- systems governed by local disturbance, scale-invariant tial for chaos, exhibit abrupt changes with no advanced power-law structures that are found for a large-scale warning in the time series [238]. As we mentioned ear- parameter range vanish as a critical transition is ap- lier in the section on multiple stable states, simplified proached [236]. In systems that have self-organized models of the system that include the essential com- regular patterns [237], particular spatial configurations ponents, interactions, and drivers and an element of may arise in the vicinity of a critical transition. Take stochasticity can be constructed [238]. Examples in- the model of desert vegetation as an example; the na- clude a minimal model of ecosystem catastrophic shits ture of pattern changes from maze-like to spots because [28], the one-dimensional grazing ecosystems model of a symmetry-breaking instability [170], as a critical [31], and the coral reef model [91]. However, these transition to a barren state is neared. models are built for low-dimensional systems neglecting the interactions between the studied components The alternative stable states separated by critical and other species. Thus, they are not applicable to high- thresholds also occur in ecosystems ranging from range- dimensional ecosystems. lands to marine systems [82, 99]. They are usually related to hysteresis so that it is difficult for a system to Recently, Gao et al. [12] developed a general analyti- recover once the system reaches state across a tipping cal framework to map the dynamics in high-dimensional point. For instance, as we mentioned earlier, in the arid systems into effective one-dimensional system’s dy- ecosystem [171], the recovery from a barren state may namics. This model can be used to accurately predicts require more rain than is needed to preserve the last the system’s response to diverse perturbations and it patches, since in such a case the concentration of soil also correctly locates the critical points, at which the water under vegetated patches already disappeared. system loses its resilience. On one hand, by using the proposed dimension-reduction method, the patterns of In summary, despite lack of universal early-warning resilience is found to depend only on the system’s intrin- signals, effectively detecting them and taking actions to sic dynamics, independent of the network topology. On push the system far away from the tipping points is very the other hand, although the changes in topology does important in ecological management. not alter the critical points, three key structural factors: 21 the system’s components, and the weighted adjacency matrix Ai j captures the rate of interactions between all pairs of components. Similar to one-dimensional systems shown in Fig.2, the resilience of multi- dimensional systems can be captured by calculating the stable fix point of equation (2.8). However, this point may depend on the changes in any of the N2 parameters of the adjacency matrix Ai j. Moreover, there are maybe different forms of perturbations bringing changes to the adjacency matrix, for example, node/link removal, weight reduction or any combination thereof. It means that the resilience of multi-dimensional systems depends on the network topology and the forms of perturbations. For large-scale multi-dimensional models, it is impossible to predict their resilience by direct calcu- lations on equation (2.8). A framework based on dimension reduction addresses this challenge. In a network, the activity of each node is governed by its nearest neighbours through the interaction term PN j=1 Ai jG(xi, x j) of equation (2.8). By using the average nearest-neighbour activity, we could get an effective Figure 22: Tipping points and ecological regime shifts are difficult to predict, since different patterns may appear even if systems are under state xeff of the system: same perturbations. (A) The blue and red ecosystems exhibit a T out change in state that tracks the incremental change in driver, but the 1 Ax hs xi x = = , (2.9) blue ecosystem provides no early warning of approaching the tipping eff 1TA1 hsi points, while the red recongizes an early warning in the form of increased variation about its mean state. (B) Both ecosystems possess where 1 is the unit vector 1T = (1, ..., 1)T; sout = relatively stable states until an abrupt disturbance occurs which out out T initially alters their states. The blue ecosystem recovers from the (s1 , ..., sN ) is the vector of outgoing degrees with disturbance and returns to its original state, while the red ecosystem out PN in in in T s j = i=1 Ai j; s = (s1 , ..., sN ) is the vector of in- is pushed beyond a tipping point and transitions to an alternate state. coming degrees; the term of the right hand of the equa- Source: Figure from [238]. out 1 PN out out in tion hs xi = N i=1 si xi; hsi = hs i = hs i is the average weighted degree. density, heterogeneity and symmetry, could affect a sys- If the adjacency matrix Ai j has little correlations, the tem’s resilience by pushing systems far from the criti- multi-dimensional problem could be reduced to an ef- cal points, enabling sustainability under large perturba- fective one-dimensional problem by using the effective tions. The study of universal resilience patterns in com- state xeff, which is plex networks suggests potential intervention strategies x f (β , x ) = eff = F(x ) + β G(x , x ), (2.10) to avoid the loss of resilience, or design principles for eff eff dt eff eff eff eff optimally resilient systems that can successfully cope with perturbations. where βeff is the nearest neighbour weighted degree that In a multi-dimensional system, the dynamics of each can be written as component is not only depend on the self-dynamics, but 1TAsin hsout sini also related to the interactions between the components βeff = = . (2.11) 1TA1 hsi and its interacting partners [242, 52]. The dynamic equation of a multi-dimensional system consisting of N Therefore, the N2 parameters of the microscopic de- components (nodes) can be formally written as scription Ai j collapses into a single macroscopic resilience parameter βeff. Any impact on the state of the dx XN i = F(x ) + A G(x , x ), (2.8) system caused by the changes in Ai j is fully accounted dt i i j i j β j=1 for by the corresponding changes in eff, indicating that the system’s resilience described by equation (2.10) is where xi(t) represents the activity of node i at time t, independent of the network topology Ai j, uniquely de- F(xi) and G(xi, x j) show the dynamical rules governing termined by the system’s dynamics F(xi) and G(xi, x j). 22 Figure 23: Network resilience in multi-dimensional systems. (A-C), The 3D plots show the resilience plane for a four-node system. (D), After applying the dimension-reduction method [12], the multi-dimensional manifold shown in (A-C) collapses into a one-dimensional resilience function in β-space (D). Source: Figure from [12].

Figure 23 shows that by mapping the multi-dimensional Taking the mutualistic networks as an system into β-space, the system’s response to diverse example, their dynamics could be cap- perturbations and tipping points can be accurately pre- tured by the following equation [12] dicted. N dx x x X xi x j i = B + x (1 − i )( i − 1) + A , (2.12) dt i i K C i j D + E x + H x i i j=1 i i i j j

where Bi is a constant influx due to migration, the sec- eters in equation (2.12) are set as node-independent: ond term defines logistic growth [243], incorporating Bi = B, Ci = C, Ki = K, Di = D, Ei = E and Hi = H. the Allee effect [244]; the interaction term captures the Mapping the multi-dimensional equation (2.12) symbiotic contribution of x j to xi, which saturates when into a one-dimensional dynamics, we could obtain the populations are large. For simplicity, the six param- 2 xeff xeff xeff f (βeff, xeff) = B + xeff(1 − )( − 1) + βeff , (2.13) K C D + (E + H)xeff

c According to the rule of stability of motion, the critical point βeff can be calculated by the following equations   x x x2  f (β , x ) = B + x (1 − eff )( eff − 1) + β eff = 0,  eff eff eff K C eff D + (E + H)x  eff (2.14)  2 2  ∂ f (βeff, xeff) x 2(C + K) (E + H)x + xeff  = −3 eff + x − 1 + β eff < 0.  eff eff 2 ∂xeff CK CK [D + (E + H)xeff]

In addition, using the equation f (βeff, xeff) = 0, we could describes βeff in function of xeff as x x D + (E + H)x β (x ) = −[B + x (1 − eff )( eff − 1)] eff , (2.15) eff eff eff K C 2 xeff

23 which is the inverse of the desired resilience function. naive way which subtracts the concerned nodes and iso- By inverting equation (2.15), i.e., swaping the axes, the lated them from the other part of the network, the au- resilience function for this system can be graphically thors use a mean field approximation to account for the obtained, which predicts a bifurcating resilience func- impact of the other part of the network, and summa- tion, and a transition from a resilient state with a single rize their impact using a resilience parameter βeff as dis- stable fixed point, xH, to a non-resilient state in which cussed above. The proposed tool can extract very close both xH and xL are stable. The critical point of this bi- approximations to the true steady state dynamics in the c furcation is predicted to be βeff = 6.79. Such value is full network (Fig. 24). In contrast, the state-of-art is fully determined by the dynamics, independent of the the naive approach, which produces completely wrong network topology Ai j. conclusions. Moreover, most real networks, especially Based on the equivalent one-dimensional model, biological and ecological networks, are incomplete, this Zhang et al. [245] derive a new centrality index: re- method can help us infer the true dynamics from these silience centrality, to quantify the ability of nodes to af- incomplete networks. Similarly, Jiang et al. [247] com- fect the resilience of the system. The resilience cen- bine the mean-field theory with combinatorial optimiza- trality of a node is mainly determined by the degree tion to infer the topological characteristics, such as de- and weighted nearest-neighbor degree of the node. This gree, from the observed incomplete networks. centrality performs better in prioritizing node’s impor- Take the fact that the mutualistic networks contain tances in maintaining system’s resilience than other cen- two different types of nodes: pollinators and plants tralities, such as degree, betweenness and closeness. into consideration, Jiang et al. [54] point out, a The proposed centrality metric enables design of effec- two-dimensional model is necessary to capture the tive strategies to protect real networks, such as mutual- bipartite and mutualistic nature. The authors use istic networks. letters P and A to denote plants and pollinators, Take advantages of the mean-field approach in and S P and S A to represent the numbers of plants dimensional-reduction, Jiang et al. [246] develop a tool and pollinators in the network, respectively, and the to learn the true steady-state of a small part of the net- model is written as the following equation [248, 249] works, without knowing the full network. Unlike the

 S P PS A (P)  dPi X γ Ak  (P) − (P) k=1 ik  = Pi αi βi j P j + P + µP ,  dt 1 + h PS A γ( )A  j=1 k=1 ik k  (2.16)  XS A PS P γ(A)P  dPi (A) (A) k=1 ik k  = Ai αi − κi − βi j A j + + µA .  dt PS P (A)  j=1 1 + h k=1 γik Pk

The notation is as follows: Pi and Ai are the abundances strength of the mutualistic interaction. of the ith plant and the ith pollinator, respectively; α By assuming that the decay parameters for all the pol- is the intrinsic growth rate; βii and βi j are the param- linators have an identical value: κi ≡ κ, and pollinators eters affecting intra-specific and interspecific competi- die from the mutualistic network one after another as tion respectively. Typically, intra-specific competition is a result of increasingly deteriorating environment, the stronger than interspecific competition, that is βii βi j. high-dimensional mutualistic network can be reduced The parameters µP and µAdescribe the immigration of to a dynamical system that contains two coupled ODEs plants and pollinators, respectively, and γ quantifies the (Ordinary Differential Equations): one for the pollinators and another for the plants, which can be written as

  dPeff 2 hγPiAeff  = αPeff − βP + Peff + µ,  dt eff 1 + hhγ iA  P eff (2.17)  dA hγ iP  eff − 2 − A eff  = αAeff βAeff κAeff + Aeff + µ, dt 1 + hhγAiPeff

24 Full Real Network

Naive High

This Paper State Value Naive High Our Method Naive Low Naive Low Figure 24: Predicting steady-state Time Time Time abundances of 5 species interacting in a (Above) Trajectories of the 5 vertices in the sampled graph G(s). (Left) Scatter plot that compares steady- state values. Our method achieves near perfect prediction on the observed 5 nodes. The naive method gets all larger 97 species ecological network. state values wrong, but even more worrying is that a single stable equilibrium bifurcates into two equilibria, Predictions use only the interactions of those and the steady-state depends on initial conditions. In the low state, all species go extinct. Naively simulating Predicted Steady State the nonlinear coupled ODE on the subgraph is a disaster. ﬁve species (incomplete information). True Steady State Source: Figure from [246].

where the dynamical variables Peff and Aeff are the 2.3.3. Control tipping points in ecosystems effective abundances of plants and pollinators, respec- The final goal of uncovering the underlying mecha- tively. The parameter α denotes the effective growth nisms behind tipping points is to develop biologically rate for the network, β describes the combined effects viable management/control principles and strategies to of intra-specific and interspecific competition, κ is the remove the tipping point to delay the occurrence of un- species decay rate in an averaging sense, and the pa- wanted critical transitions across different scales [239]. rameter µ accounts for the migration effects for the These transitions include global ecological state transi- species. Two effective mutualistic interaction strengths, tion at the planetary scale [157], the shutdown of the hγPi and hγAi, can be obtained by averaging as: (i) thermohaline circulation in the North Atlantic at the re- unweighted average, (ii) degree-weighted average, and gional scale [253], the switch of shallow lakes from (iii) eigenvector-based average. Figure 25 shows the clear to turbid waters and global extinction of species ensemble-averaged pollinator and plant abundances in at local scales [96, 199, 254]. Due to the nonlinear na- networks obtained by different averaging ways. The re- ture of ecosystems, it is difficult to control them since duced model through averaging method (ii) and (iii) has controlling nonlinear dynamical networks remains to be a remarkable predictive power for the tipping point. an outstanding problem and is currently an active area Jiang et al. [54] use 59 empirical mutualistic net- of research [254, 255]. Especially for the tipping point works extracted from real data to show the reduced two- control problem, Nishikawa et al. [256, 257, 258] in- dimensional model’s power of predicting tipping points vestigate how small perturbations can be used to drive where systems lose their resilience. In a system, the the system to the desired attractor. Vidiella et al. resilience is usually reflected by the relationship be- [259] demonstrate that in semiarid ecosystems, the phe- tween the average species abundance and some param- nomenon of an ecological ‘ghost,’ a long transient phase eter with variations that reflect the impact on the envi- during which the system maintains its stability, may be ronment caused by stressors, such as global warming exploited to delay or prevent the occurrence of a tipping or overuse of pesticides [250, 251, 252]. In mutualis- point [259]. tic networks, there are three important parameters re- A tipping point transition is the consequence of grad- lated to the system’s resilience: (i) fn—the fraction of ual changes in the system caused by a slow drift in the pollinators that have become extinct because of envi- intrinsic parameters and/or external conditions. The in- ronmental deterioration; (ii) κ—the average pollinator trinsic parameters include the species decay rate, the decay rate; and (iii) fl—the fraction of links destroyed mutualistic interaction strength, and the fraction of disas a result of the death of a fraction fn of pollinators. appeared nodes and/or links in an ecological network, Jiang et al. [54] calculate the average abundances of which could be altered by environmental conditions plants and pollinators under the variations of these three [239]. Thus these parameters are also called ‘environ- parameter. The authors find that for resilience functions mental parameters’, and ecological system states can be without tipping points, the abundance variations in two- controlled by changing these parameters. For instance, dimensional reduced model through unweighted aver- a sudden bloom of cyanobacteria in a lake or a reservoir age are in good agreement with the original systems. can be devastating because it kills fish on a large scale For the resilience functions with tipping points, the re- and poses great toxicity risks for the environment. The duced model through degree or eigenvector weighted effective way to prevent a bloom of cyanobacteria in a average has a remarkable predictive power for tipping lake or a reservoir is to stop or significantly reduce nu- points, even in the presence of stochastic disturbances. trient inflow into the lake [260]. Another example is the 25 3 3 A 2.5 2.5 B

2 2

eff 1.5 eff 1.5 A P SH SH 1 UWAH 1 UWAH DWAH DWAH 0.5 EWAH 0.5 EWAH L L 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f f n l

Figure 25: Resilience functions with tipping points in networks constructed from the data recorded at Tenerife, Canary Islands [262]. (A), Ensemble-averaged pollinator abundance with high initial values versus fn, which is the fraction of removed pollinators. (B), Ensemble-averaged plant abundance with high initial values versus fl, the fraction of removed mutualistic links. Here the parameters are (A) (P)=1 (A) (P) h = 0.2, t = 0.5, βii = βii , αi = αi = −0.3, µA = µP = 0.0001, γ0 = 1 and κ = 0. Reproduced from Ref. [54].

Figure 26: Managing a tipping point caused by an increase in the fisheries food web. If we could detect adequate early- pollinator decay rate. The management principle is to maintain the warning signs in time, the regime shifts can be averted abundance of the pollinator with the largest number of mutualistic by stipulating a rapid reduction in angling and/or intro- connections (the horizontal light blue dashed line). Without ducing gradual restoration of the shoreline [261]. abundance management, all pollinator populations collapse abruptly at a single value of κ—a tipping point. However, with abundance Very recently, Jiang et al. [239] investigate how to management, the extinction process becomes gradual, effectively manage or control tipping points in real-world complex removing the tipping point. Figure from Ref. [239]. and nonlinear dynamical networks in ecology by altering the way that species extinction occurs. An example could be replacing massive extinction of all species before the collapse. However, with abundance manage- by gradual extinction of individual species, as the envi- ment the hysteresis loop disappears and species recov- ronmental parameter continues to increase, so that the ery begins at the point of global extinction. Especially, occurrence of global extinction is substantially delayed. when the the environmental parameter is the mutualis- The authors focus on a large number of empirical pol- tic interaction strength, species recovery is not possi- linator–plant bipartite networks, whose dynamics are ble without the abundance management, but a full re- governed by mutualistic interactions [249, 54]. These covery can be achieved with it [239]. In addition, the networks are managed by choosing a ‘targeted’ species species recovery point can be predicted reasonably well and maintaining its abundance (or keep the decay rate of by a two-dimensional reduced model [54] derived un- this species unchanged) as the environmental parameter der the condition that abundance control/management is is increased. The abundance management in such ways present. The management or control strategy to main- can remove the tipping point and delay the occurrence tain the abundance of certain pollinator species may be of total extinction, as shown in Fig. 26. The amount realized through the approach of injecting robotic pol- of delay depends on the particular species chosen as the linators [264] to expedite recovery [265], which may target. All species can be ranked by the amount of de- help address the devastating problem of relatively sud- lay they induce, which characterize the control efficacy. den disappearances of bee colonies, which are happen- Their ranking is found to be determined solely by net- ing currently all over the world. work structure with no relations to the intrinsic network Except these studies on preventing or delaying tip- dynamics [263]. ping points, another branch of studies related to re- In the absence of abundance management, a hystere- silience management focus on how to recover or rebuild sis loop arises when attempts to restore the species pop- ecosystems that is already in its unwanted state or prone ulation are made by improving the environment, i.e., by to tipping points [267, 266, 268, 269, 270]. Take the making the environmental parameter change in the di- system of coral reefs as an example, which was in a rection opposite to the one that led to an extinction. In pristine condition in the pre-industrial time period. Now such case, in order to revive the species abundances to many coral-reef ecosystems have been degraded to re- the original level, the environmental parameter needs to silient assemblages no longer dominated by live coral, be further away from the tipping point, i.e., the envi- and reversing this degradation will require a reduction ronment needs to be significantly more favourable than in human pressures on reefs and improved management 26 propriate policies, it is also important to act early, since once a system has crossed a threshold, transition to a new stable state may take many years. Moreover, it has been found that exceptional weather events, such as hurricane, may also create windows of opportunity for phase-shift reversals on coral reefs [266]. If we could restrict fishing [269], establish networks of herbivore management areas [270], and take advantages of shocks, it may be easier to rebuild a coral-dominated state in coral reefs. Since all the systems in the planetary scale have become ever more networked and interdependent, due to human interventions, there is a growing need to focus on managing the resilience in ecosystems worldwide. Another thing to bear in mind is that all ecosystems are currently under rapidly changing environment. It is important to embrace the novel conditions and propose realistic goals for resilience management. Thus, adaptive governance [271] has been suggested as a suitable approach. It rests on the assumption that landscapes and seascapes need to be understood and governed as complex social-ecological systems rather than as ecosys- Figure 27: Alternate states in coral reefs in different time period. The tems alone. Next we will review literatures on phase condition and composition of the coral-dominated state has changed through time, away from a pristine state (dashed vertical line above transitions in biological networks. “coral”). From the (A) pre-industrial through (B) early industrial to (C) present, when the coral-dominated reef state has become less common, and could become uncommon in the future (D) under a 3. Phase transitions in biological networks “business as usual” scenario. However, (E) if appropriate policies are implemented, more reefs may be maintained in or shifted back to In a living organism, there are many biochemical pro- coral-dominated states, with a reduction of the resilience of the cesses, forming various forms of biological networks, algal-dominated state and increased resilience of the coral-dominated state. Reproduced from Ref. [266]. such as gene regulatory networks, protein interaction networks, and metabolic networks at the molecular level [47, 272], cell interaction networks at the cellular level [273], and disease networks [274] at the phenotypic of ecosystem processes that weaken the degraded con- level. All these biological networks evolve dynami- dition and promote corals [266]. The resilience of reef cally and may have strong nonlinearity inside. Their states can be expressed as deep or shallow valleys in a responses to internal signals or external perturbations stability landscape, as shown in Fig. 27. Deeper valleys are usually not gradual but show switch-like behaviors indicate higher levels of resilience, whereas shallow val- [275]. The authors state the biological network shifts leys are indicative of low resilience. As natural and from one state to another abruptly, such as cell-fate in- anthropogenic drivers have changed reef systems, the duction [276] or the onset of diseases or cancer [277]. coral-dominated state has become less resilient, while Such state transitions can be modeled by the stabil- the algal-dominated state has become more resilient. If ity landscape [28] and analyzed by using the resilience we let the business go as usual without management, framework from ecology [4]. As discussed in Chapter very few reefs will be maintained in a coral-dominated 2, many abrupt state shifts are usually associated with state. If appropriate policies are implemented, more multiple stable steady states [56]. In the following we reefs can be sustained or returned to a coral-dominated will review under what conditions biological networks (but non-pristine) state, providing society with critical exhibit multiple stable steady states. goods and services. Yet, to recover or rebuild the community composition of coral reefs is very difficult. It require scientists, policy makers, managers and resource 3.1. Bistability in biological systems users to act collectively to develop long-term commit- Similar to the “multiple stable states” in ecological ments to improve reef management. In addition to ap- systems as discussed in Chapter2, a system-level prop- 27 erty called “bistability” (or, more generally, multistabil- removed. ity) universally exists in all biological systems [278], Feedback itself alone does not guarantee that a sys- and it may be of particular relevance to biological system will be bistable. A bistable system must also have tems that switch between discrete, alternative stable some type of non-linearity within the feedback circuit states; generate oscillatory responses; or “remember” [281]. That is, the enzymes in the feedback circuit must transitory stimuli [279]. Uncovering the mechanisms respond to their upstream regulators in an ‘ultrasensi- underlying “bistability” is crucial for understanding ba- tive’ manner, which converts continuously varying sig- sic cellular and biochemical processes, such as cell cy- nals into discrete outputs (ON or OFF responses) [292]. cle progression, cellular differentiation, cellular apopto- A bistable circuit would always exhibit some degree of sis [57], and the onset of disease and cancer, as well as hysteresis, indicating that the system has alternative sta- in the origin of new species [280]. Due to the prevalence ble states under certain stimulus. The response to stimu- and fundamental importance of bistability in biochemi- lus is related to the system’s previous state, and it will be cal systems, plenty of theoretical and experimental stud- harder to flip the system from one state to the other than ies have been carried out to uncover what is needed it is to maintain the system in its flipped state [281], as for a signal transduction pathway to exhibit bistabil- shown in Fig. 28A. In such case, the bistable switch is ity [281, 282, 283], which is a emergent phenomena two-way (it can be switch back and forth between ON in networks of biochemical reactions rather than single and OFF states), which are characteristic of metabolic molecules or single reactions. pathways, such as lac operon [290]. If the feedback in a circuit is strong enough, the system may exhibit irre- 3.1.1. Generators of biological bistability versibility (Fig. 28B). The system could stay in the the In cell signaling [284], most (or perhaps all) of the flipped state indefinitely after the triggering stimulus is biochemical reactions are reversible. For instance, there removed. In such case, the bistable switch is one-way are DNA methylation and DNA demethylation, and pro- (once flipped, it cannot be turned back), which play ma- teins are phosphorylated and dephosphorylated. How- jor roles in developmental transitions, such as apoptosis. ever, many biological transitions are essentially irre- Even feedback regulation has been considered to be versible, such as cell differentiation [285, 286, 287], a prerequisite for bistable behavior [287, 291]. Yet, cell cycle, and immune stress response [288]. A cru- it is not a necessary component of switching phenom- cial question has been raised: how might the reversible ena. For example, in biochemical reaction networks, the reactions lead to practically irreversible changes in cell sources of bistability can lurk behind the fine mechanis- fate [281]? tic details of even a single overall reaction [293, 294, Monod and Jacob [289] proposed that the answer 295]. For example, in protein kinase cascades, bistabil- lay in the way that gene regulatory systems are wired, ity and hysteresis can arise solely from a distributive ki- where feedback loops are mechanisms required for pro- netic mechanism of the two-site Mitogen-activated producing biological bistability and irreversibility. There tein kinase (MAPK) phosphorylation and dephosphory- are different types of signal transduction circuits that lation, with no apparent feedback [296]. In summary, could achieve such goal, such as pairs of natural tran- epigenetic bistability appears to be at the heart of de- scriptional repressors wired to inhibit one another [290, cisive, irreversible biological state transitions, such as 291], and a positive feedback loop composed by activa- cell differentiation and cell cycle [297, 298, 299, 300], tors sharing the same opposing repressors in gene reg- which arises from feedback loops and ultrasensitivity ulation. For example, let’s consider two gene products, [297, 280] or particular network topology with a certain P1 and P2, each of which inhibits the other’s transcrip- range of parameters [294, 301]. tion, thus the system has a stable state with P1 on and P2 off, or an alternative stable state with P1 off and P2 on. Once either stable state has been established, the system 3.1.2. Mathematical models of biological bistability could stay in such state until external stimulus push the Computational modeling and the theory of nonlinear system to transition to the alternative stable state. Sim- dynamical systems allow one not only to describe the ilar behavior presented in positive feedback loops (mu- bistability, but also to understand why it occurs [302]. tual activation toggle-switch motifs [290]), the system Next we will review two claissical mathematical models would toggle back and forth between a state with both developed to show how bistability arises [303, 278, 304, A and B off and a state with both A and B on [281], 305]. showing an ability of ‘remembering’ a transient differ- A simple positive feedback loop (a one-ODE model). entiation stimulus long after the triggering stimulus was As discussed above, a most common mechanism for 28 quires a sigmoidal shape but still monostable shown in Fig. 29C and 29D. If f keeps increasing, the system shows hysteresis and becomes bistable for some values of stimulus, as shown in Fig. 29E. Eventually, the feedback becomes so strong that the system show essentially irreversible response, as shown in Fig. 29F. This simple model has been successfully used to explain the bistability or irreversibility in many biological processes, such as the maturation of Xenopus oocytes [303], induced osteogenic differentiation in a myogenic subclone [287], and the white–opaque switch in Candida albicans [309]. A mutually inhibitory network (a two-ODE model). As we mentioned earlier, in addition to positive feed- Figure 28: Hysteresis and irreversibility in bistable signaling circuits. back loops, a double-negative feedback loop, also called (A) Any bistable circuit should exhibit some degree of hysteresis, mutually inhibitory network, can generate bistability meaning that different stimulus/response curves are obtained [281]. It consists of two signaling proteins, repressor depending upon the system’s previous state. (B) Irreversibility is achieved when a bistable system has very strong feedback. U and repressor V, which inhibit each other’s expres- Source: Figure is modified from [281]. sion or activation, as shown in Fig. 30A. Given certain stimulus, the system could switch between two distinct states: one state with high expression level of repressor generating bistability is the existence of positive feed- U and low expression level of repressor V, another with back loops, which directs two mutually exclusive cell low expression level of V and high expression level of states [306, 307]. A positive feedback loop may be U. Its nonlinear dynamics could be captured by the fol- formed from one or two signaling proteins [308], and lowing two-ODE model, which is also called the ‘toggle the simplest form consists of one single signaling pro- switch’ model [278, 310]: tein that can be reversibly switched between an inac- dU α tive form (A) and an active one (A*). As shown in Fig. = 1 − d U dt 1 + k Vβ 1 29A, the transition between A and A* that is assumed 1 (3.2) dV α to be regulated by an external stimulus and the positive 2 − = γ d2V feedback between the transition and the stimulus. This dt 1 + k2U process can be modeled by the following ordinary dif- where the first terms represent the effect of negative ferential equation (ODE) [303]: feedback loops, and the second terms denote degrada- d[A∗] tion/dilution of the repressors. In the model, variables ={stimulus × ([A ] − [A∗])} dt tot U and V represent the concentration of two repressors; (3.1) [A∗]n parameters α1 and α2 respectively represent the max- + f − kinact[A∗] imal production rate of repressors U and V; β and γ Kn + [A∗]n are cooperativity coefficients of repressors V and U, re- where the first term represents the basal transcription spectively; parameters k1 and k2 describe the repression rate due to external stimulus; the second term denotes strengths, which is determined by the binding rate and effect of the positive feedback that is modeled by a non- dissociation rate of repressions to their promoters; and linear Hill equation, with K being the effector concen- parameters d1 and d2 represent the degradation rates of tration needed for half-maximum response and n denot- repressors. ing the Hill coefficient, and the parameter f represents The equilibrium solutions of this model can be found the strength of the feedback; the last term represents the by drawing the nullclines (dU/dt = 0 and dV/dt = 0). If inactivation of A* with a degradation rate kinact. the system has balanced rates of synthesis of the two re- The shape of nonlinear stimulus–response of Eq. 3.1 pressors and the cooperativity coefficients β, γ > 1, the depends on the value of f (the strength of feedback). nullclines show sigmoidal shapes and intersect at three When f = 0, it means that there is no feedback in points, exhibiting bistability with one unstable and two this system, and the response is a monostable smooth symmetrical stable steady states shown in Fig. 30B. Michaelian curve, as shown in Fig. 29B. As the feed- However, if the rates of synthesis are imbalanced, the back strength f increases, more nonlinearity, or ultra- nullclines will intersect only once, producing a monos- sensitivity, is introduced, and the stimulus–response ac- table state (Fig. 30C). The slopes of the bifurcation lines 29 Figure 29: A simple positive feedback loop and its stimulus–response curve for different feedback strength. (A) The genetic circuit of positive feedback loop. (B-F) The stimulus–response curve calculated numerically from Eq. 3.1 under different feedback strength f , with other parameters being set as: n = 5, K = 1 and kinact = 0.01. Source: Figure from [303]. are determined by the cooperativity coefficients β and γ. The size of bistable region increases as the rate of repressor synthesis (α1 and α2) are increased (Fig. 30D), and reduced when β and γ are decreased (Fig. 30E). Note that β and γ represent the degree of cooperativity in the binding. The case β, γ = 1 corresponds to a linear repression and the uncooperative binding of monomers, while β, γ > 1 requires the cooperative binding of two or more repressor proteins [311] that need to form polymer or multiple repressors so as to cooperatively bind to promoters that own more than one operator sites [310]. Thus, cooperative binding is a necessary condition for the double negative feedback loops to show alternative stable states. The mutually inhibitory network have been widely used to describe the bistable phenomenon that exists in the growth and development of organisms. For instance, in mammalian embryonic stem cells, the decision between the epiblast and primitive endoderm fate can be described by a simple mutual repression circuit modulated by FGF/MAPK signaling [312]. The lysogeny–lytic bistable switch in bacteriophage λ can be described by similar models [313, 314, 315, 316]. In addition, such model can be used to engineer artificial gene networks in mammalian cells, and the devel- Figure 30: Geometric structure of the two-ODE model. (A), oped epigenetic circuitry is able to switch between two Schematic illustration of the mutually repression network. (B), One unstable and two stable states appear with balanced rates of synthesis stable transgene expression states after transient admin- of the two repressors. (C), A monostable states arise with unbalanced istration of two alternate drugs [317], enabling precise rates of synthesis of the two repressors. (D) The bistable region. The and timely molecular interventions in gene therapy. lines mark the transition (bifurcation) between bistability and monostability. The slopes of the bifurcation lines are determined by Besides the two basic math models reviewed above, the exponents β and γ for large α1 and α2. (E) Reducing the plenty of derived models have been proposed to de- cooperativity of repression (β and γ) reduces the size of the bistable scribe the bistability or irreversibility in various bio- region. Other parameters are set to k1 = k2 = 1 and d1 = d2 = 1 logical processes. For example, the bistability in p54 Source: Figure is modified from [278]. steady states can be explained by a positive feedback loop composed of transcriptional factor p53 and a mi- croRNA, miR-34a, where p53 upregulates the transcrip- eled by a two-dimensional ODE model [305], as shown tion of miR-34a, and in turn the miRNA indirectly up- in Fig. 31. Martinez et al. [319] build a minimal delay- regulates p53 expression via repressing SIRT1, a nega- differential equation (DDE) model that has a region of tive regulator of p53 [318]. Such process can be mod- bistability, which makes the subtilis biofilms jump from 30 Figure 31: Bistability in p53/miR-34a feedback loop with a mathematical model. Drawing the nullclines of two equations, one unstable and two stable states appear showing bistability. The middle bottom plot shows the evolution of p53 (the red line) and S (the green line) over time, and the bifurcation plot shows different steady states of p53 (p53ss) against different intensities of S. Source: Figure from [305]. a stable steady state to a growth oscillatory attractor ble steady states and showing hysteresis, as shown in under perturbations. Wang et al. [287] use a bistable Fig. 32. The introduce of bistability ensures that the switch model to analyze the observed differentiation Cdc2/APC system produce sustained oscillations with- behavior of human marrow stromal cells. Bala et al. out approaching a stable steady-state. Next, we will [320] propose a modification to an existing mathemati- show how the bistable switches govern mitotic control cal model of mitosis-promoting factor control in Xeno- in cell cycle. pus oocyte extract, and use MPF as a bifurcation pa- Cdc2 (also referred to as Cdk1) and cyclins form rameter, giving rise to bistability in the MPF activation a stoichiometric complex and play a key role in the module. All these theoretical studies of biological bista- control of the G2/M transition of the cell cycle [326]. bility advance our understanding of the possible under- The positive feedback in the Cdk1 activation loop and lying mechanisms behind important cellular processes, the major Cdk1-counteracting phosphatase, PP2A:B55, such as cell cycle and cell differentiation. Next, we re- feedback regulation work as bistable switches. They view empirical studies of the bistability in the growth generate different thresholds for the transition between and development of cells. different cell-cycle states: interphase and M phase [327, 328]. As shown in Fig. 33, there are distinct 3.1.3. Empirical studies of biological bistability thresholds for mitotic entry and mitotic exit, provid- Bistable switches are sufficient to encode more than ing robustness of the M phase state and preventing cells two cell states without rewiring the circuitry [321], from flipping back to the interphase state in the noisy which has been experimentally found to be at the core cellular environment [329]. In addition, the two dis- of determining the state transitions in cell cycle and cell tinct states: interphase and M phase are created by differentiation. bistable switches [330, 331], and they could be stabi- Bistability in cell cycle The cell cycles of embryos in lized by positive feedback loops without allowing the their early stage provide an example of a robust biolog- cell to come to rest in intermediate transitional states ical oscillator [322]. The sustained oscillations are gen- [332]. erated by the interlinked negative and positive feedback Bistability in cell fate switch Cell fate decision-making loops [323]. In the essential negative feedback loop, the is the process that a cell committing to a differentiated cell-division cycle protein kinase, Cdc2, activates the state in growth and development undergoes. In this pro- anaphase-promoting complex (APC), leading to cyclin cess, the bistable switch mechanism is prevalent in di- destruction and Cdc2 inactivation [322]. Under some recting two mutually exclusive cell fates [279, 287, 333, circumstances, a long negative-feedback loop by itself 306, 321]. One typical example is the maturation of is sufficient to produce oscillations [324], However, the Xenopus oocytes, where the immature oocyte represents negative feedback loop in the Cdc2/APC system alone a default fate and the mature oocyte represents an in- may produce damped oscillations [322]. Pomerening et duced fate [334]. At a biochemical level, oocyte matu- al. [322] experimentally show that the positive feedback ration is controlled by p42 MAPK and cyclin B/Cdc2, loops in the Cdc2/APC system. It consists of Cdc2- which are known to be organized into positive feed- mediated activation of Cdc25 and inactivation of Wee1 back loops [281, 303]. For example, Mos activates p42 and Myt1, which could function as a bistable system MAPK through the intermediacy of MEK. Once ac- [325] by toggling between two discrete alternative sta- tive, p42 MAPK feeds back to promote the accumula- 31 AB C D

Figure 32: Three ways that Cdc2 respond to diﬀerent concentrations of non-degradable cyclin. (A) The Michaelian response would be expected if cyclin had directly activated Cdc2. (B) The ultrasensitive response could arise from multistep activation mechanisms, from either stoichiometric inhibitors or from saturation eﬀects. (C) The bistable response could arise from a combination of ultrasensitivity and positive feedback. Source: Figure from [297].

Figure 33: Bistable Switches of Mitotic Control. Schematic signal-response diagram for Cdk1 auto-activation (A), PP2A:B55 feedback regulation and (B), mitotic substrate phosphorylation by interlinked kinase–phosphatase switches (C). Source: Figure from [329].

32 tion of Mos [335]. Such positive feedback loop makes backs, leading to a very low rate of differentiation. p42 MAPK show an all-or-none, bistable response to In summary, it is generally considered that the progesterone or microinjected Mos [336]. Xiong et progress of biological processes is controlled by mech- al. [303] provides experimental evidence that the p42 anism of bistablity. This mechanism has been discov- MAPK and Cdc2 system with the strong positive feed- ered in cell differentiation [285, 286, 287], lysogeny- back loops produce bistability and generate irreversible lytic switch in bacteriophage λ [314], Immune stress biochemical response from a transient stimulus [303]. response [288], the lactose utilization network in Es- Another example of bistability in cell differentia- cherichia coli [339] and so on. The phenomenon of tion that has been experimentally verified is the classic bistable allows the cells generate a response in a digital bistable bacteriophage λ switch [337]. This switch is manner, that is discretely switch between two or more composed of two mutually repressive TFs, CI and Cro, distinct stable states. Exploring the bistable responses and their expression level determine cell-fate develop- in organism help us to understand the inherent mech- mental decision-making, as it reproduction infects its anism of corresponding processes and further advance host [338]. The expression of CI but not Cro confers our studies of biomolecular controller [342, 343]. lysogenic growth, and the expression of Cro but not CI confers lytic growth [321]. Fang et al. [321] demon- 3.2. Resilience at multiple levels of biological network strate experimentally the emergence of two new expres- Resilience in a biological network is characterized by sion states in the model of bistable switch of the bac- its ability to maintain functions and adjust its dynamics teriophage λ. They constructed strains XF204, XF214 to insure a consistent output, when it is exposed to both and XF224, where the expression level of Cro is reduced external and internal perturbations. Accurately predict- in the order [Cro]XF224 < [Cro]XF214 < [Cro]XF204. This ing the output of a dynamical biological network with reduction quantifies the expression levels of CI and Cro respect to perturbations is crucial for understanding sig- in individual cells of the three strains at different tem- nal fidelity in natural networks and for designing noise- peratures for more than 20 generations. As shown in tolerant gene networks [344]. Next, we will review the Fig. 34, the typical bistable behavior appears. Cells pre- studies on resilience in biological networks at multiple dominately exist in a low-Cro and high-CI levels (red levels ranging from simple generic circuits to complex lines) at a low temperature. They flip to a high-Cro and multicellular organisms. low-CI level (green lines) at high temperature. Between the two extreme, cells with high-Cro and high-CI levels 3.2.1. Resilience of genetic circuits in presence of (black lines) also appear, due to the different speeds of molecular noises the switching the expression levels of Cro and CI [321]. Both the theoretical and empirical studies of biolog- Besides the experimental studies of cell fate switches ical bistability reviewed in Sec. 3.1 are nearly all de- in unicellular organism or virus, plenty of studies show terministic. Here is no consideration of the common that bistability commonly exist in cell differentiation presence of stochastic fluctuations, which are induced of multicellutar organism. For example, Wang et al. by extrinsic and intrinsic noises [345]. The former [287] use a human bone marrow stromal cell subclone are communicated by exogenous sources, such as os- to study myogenic and osteogenic differentiation. They cillatory cascades that regulate progression of cell cy- that BMP2-induced osteogenic differentiation of these cle [346, 347] or environmental stressors [348]. The cells exhibits a threshold effect and an all-or-none re- intrinsic noises can be interpreted as random fluctua- sponse, which can be successfully analyzed by using tions within an individual cell. Typically, they alter the the bistable switch model [339]. Bhattacharya et al. intensity of a signal, leading to an altered stoichiomet- [288] show that two mutually repressive feedback loops ric relationship between the input and the output sig- can generate a bistable switch capable of directing B nals [345]. Such fluctuations are inherent properties cells to differentiate into plasma cells. In this process, of transcriptional, post-transcriptional and translational it is important the differentiated cells are able to exe- dynamics [349]. For example, replication-transcription cute certain biological functions that requires that they conflict [350] and RNA polymerase backtracking me- have a low de-differentiate rate [340]. The underly- diated by R-loop formation [351] are stochastic events ing mechanism that prevent differentiated cells from de- that invoke intrinsic noise in an individual cell [347]. differentiating is the positive feedback in cell signaling The intrinsic noise, or molecular noise, commonly ex- pathways. Ahrends et al. [341] show that an ultrahigh ists in cells and enables the phenotypic diversification feedback connectivity exists in pre-adipocytes of mam- of completely identical cells exposed to the same envi- malian tissues, supporting more than six positive feed- ronment [349]. 33 A

Figure 34: Expression levels of CI and Cro in strains XF204, XF214, and XF224 at different temperatures showed more than two expected cell populations. (A) Percentages of cells having CI only (red), Cro only (green), or both CI and Cro (black) in strains XF204, XF214, and XF224 at different temperatures. (B) Representative fluorescent images of XF224 cells B showing CI expression (yellow polelocalized Tsr-Venus-Ub spots) and Cro expression (yellow quarter/midcell-localized LacI-Venus-Ub spots) at low, intermediate, and high temperatures overlaid with phase-contrast cell images (gray). Source: Figure from [321].

The inherent stochasticity of biochemical processes is network from properly functioning and limits the ability inevitable. It arises due to the random nature of chem- to biochemically control cellular dynamics [344, 357]. ical reactions within a cell [352]. When only a few The good side of noise can only show in the large evo- molecules of a specific type exist in a cell, stochas- lutionary time scale and in large biological populations, tic effect can become prominent [345]. For example, while its bad side influences the molecular activities in the feedback loops (toggle switch) that generate bista- cells all the time. Life in the cell is a complex battle bility reviewed in the previous section, can randomly between randomizing and correcting statistical forces, switch between two states in the presence of noise and many control circuits have evolved to eliminate, [278, 353, 354, 355, 356]. Furthermore, the fluctuation tolerate or exploit the noise [358, 359, 360, 361, 357]. in small systems, especially with low molecular con- For example, negative feedbacks could suppresses noise centrations, can lead to additional states which are es- [359] and positive feedbacks could stabilized differenti- sentially unstable [310]. Example of such instability is ated states in cells [349]. provided by the empirical study of the classic bistable Despite the presence of molecular noises, the switch- bacteriophage λ switch [321] reviewed above. The like behavior in biological networks would not be de- switch based on the expression levels of two mutually stroyed. For example, bistable switches in the pro- repressive transcription factors (CI and Cro) not only tein interaction network can operate reliably against the stay in two stable states as (low, high) and (high, low), stochastic effects of molecular noise [290]. This is be- but also show two unstable states as (high, high) and cause circuits consisting of transcription factors and mi- (low, low) with different probabilities. Such stochastic croRNAs can not only show biological bistability as dynamic behavior of gene regulatory network is gov- discussed in Sec. 3.1, but also confer resilience to bi- erned by a chemical master equation, which describe ological processes against intrinsic and extrinsic noise the time evolution of the probability distribution of the [362, 305]. This is accomplished through noise buffer- system state [307]. As shown in Fig. 35, a simple ing [363], which employs microRNA-based mechanism gene regulatory motif that demonstrates deterministic that keeps gene expression stable and hence decreases bistability and hysteresis without considering molecu- the variation in gene expression [305]. In addition, a lar noises. However, stochastic hysteresis loops with circadian network can oscillate reliably in the presence multiple mean states appear with different probabilities, of stochastic biochemical noise. When cellular condi- which can be simulated by different initial conditions in tions are altered, the ability to resist such perturbations the form of Gaussian distributions N(µ, σ) with mean µ imposes strict constraints on the oscillation mechanisms and standard deviation σ. underlying circadian periodicity in vivo [364]. The roles of molecular noises in biological networks The resilience in genetic circuits against stochastic can be the “two sides of a coin” [351]. On one side, noises mainly relies on the underlying connections be- the induction or amplification of genetic noise is an im- tween molecules, forming a complex genetic network portant evolutionary pro-survival strategy in unicellular that contains multiple nested feedback loops. For ex- and multicellular organisms, which can foster pheno- ample, on induction of cell differentiation, the nested typic heterogeneity in a population [347]. On the other feedback loops prevent the established phenotypes to be side, stochastic fluctuations may hinder the biological reversed even in the presence of significant fluctuations 34 A B Figure 35: Hysteresis in deterministic versus stochastic descriptions. (A). Hysteresis loop of the deterministic self-regulatory system without considering molecular noises. (B). Transient hysteresis in the stochastic self-regulatory system. Variable τ is associated to the time scale of the protein degradation. Slow transients lead to multiple mean states resulting in a transitory hysteretic behavior. Source: Figure from [307].

[365]. Two positive feedback loops are involved, one is examples of resilience in unicellular organisms under mediated by the cytoplasmic signal transducer Gal3p to environmental stress. generate bistability in cell differentiation, while the par- Cooperative growth of budding yeast in sucrose. The allel loop mediated by the galactose transporter Gal2p Budding yeast, Saccharomyces cerevisiaes, is a single- increases the expression difference between the two celled eukaryotic fungi, who uses oxygen to release the states, enhancing the induced cell differentiation state. energy from sugar. The sugar concentration could affect Another example is the circadian oscillation, which is the rate of yeast growth: up to a certain point, the higher generated by the negative regulation exerted by a protein sugar concentration results in the faster its growth (yeast on the expression of its own gene [366]. The intercel- cannot grow in very strong sugar, but this is beyond lular coupling could increase its resilience with respect the scope of our discussion here) [373]. In an experi- to molecular noise [367]. In addition, the cooperativ- ment, laboratory populations of budding yeast grow in ity in repression discussed in Sec. 3.1.1 coupled with sucrose, and a daily dilution was performed [30], where transcriptional delay could enhance biological system’s a fraction (for example, 1 in 500 for a dilution factor resilience [367, 368]. Transcriptional delay is another 500) of cells were transferred into fresh media. As the intrinsic property of genetic circuits resulting from the dilution factor increases (the sucrose concentration de- sequential nature of protein synthesis and the time that creases), yeast populations decreases and collapse at a transcription factors take to move to their target promot- tipping point, showing an abrupt phase shift.The tipping ers [369, 370]. Gupta et al. [368] find that increasing point the cooperative growth of yeast cells, which cre- delay dramatically increases the mean residence times ates a positive feedback. near stable states, and therefore stabilizes bistable gene networks. The budding yeast eats sucrose, but needs to break it down into glucose and fructose before it can get the food through its cell wall. To break the sucrose down, 3.2.2. Resilience of unicellular organisms under envi- yeast secretes an enzyme known as invertase [374]. The ronmental stress sucrose is hydrolyzed into glucose and fructose in an ex- Resilience in unicellular organisms, such as yeast, tracellular process, which can be shared between yeast cyanobacteria and other microbes refers their ability to cells, so that they can work cooperatively. Such coop- survive a disturbance [371]. It can be achieved either eration improve the cell survival in yeast populations through absorbing effects of a disturbance without a no- by efficiently processing sucrose concentration [375]. table change, or through cooperative growth so as to re- In the experiment where yeast populations grew in su- cover the abundance of its community. The ability to crose and daily dilutions with certain dilution factors tolerate a disturbance primarily depends on the traits as- were performed, bistability appears in the population sociated with individual cells, while the ability to re- densities of yeast cultures with a wide range of initial cover mainly depends on traits associated with the pop- cell densities [254]. As shown in Fig. 36, cultures ulations. Note that, in microbiology, the former insen- starting below a critical density went extinct, whereas sitivity to a disturbance is defined as resistance, and the cultures starting at higher initial densities survived and latter ability to recover is called resilience [372]. Since reached a finite stable fixed point. A fixed point is rec- that the term ‘resilience’ was first proposed to include ognized when the ratio of population densities between both abilities to withstand and recover from perturba- subsequent days nt+1/nt = 1, where nt is the population tions [4] (see Chapter2), so we use resilience to refer to density at day t (t = 1, 2, ..., 6). For the cultures start- both of these two abilities. Next, we review two typical ing near the critical density (unstable fixed point), some 35 populations survives and others go extinct due to coop- light, and negative phototaxis is movement away from erative growth. The cooperations between yeast cells light. If the light is strong and extensive, and there break down sucrose more efficiently, and more cells is no place that cyanobacteria could move away from grow, leading to a positive feedback so that the popu- the light, then the cyanobacteria will shade one another. lation could survive at a nonzero stable fixed point. In Mutual shading can ameliorate this stress and promote contrast, if not enough cooperations form at the begin- the growth of cyanobacteria, which will in turn encour- ning, a weak positive feedback could drive the popula- age more mutual shading, forming a positive feedback tion to collapse. [380]. Such positive feedback leads to alternative stable For more efficient cooperations, budding yeasts ag- states in the cell density of cyanobacteria populations gregate together building communities, or grow by in- (Fig. 37). A low population density does not provide complete cell separation forming undifferentiated multi- sufficient shading to protect itself against photoinhibi- cellular clumps. Cells in clumps of cooperate to collect tion. Hence, the population goes extinct. However, pop- food, and have a growth advantage over an equal num- ulation density above a threshold, allows the population ber of single cells in low sucrose concentrations. Cell to create sufficiently turbid conditions to suppress pho- clumps could grow when sucrose is scarce, whereas sin- toinhibition, so that the population can establish itself gle cells cannot. In addition, clumps with more cells [382]. grow faster than smaller clumps [374], showing a pos- Similar to the formation of cell clumps in yeast popu- sible origin of multicellularity and advantages of evolv- lation discussed above, the cyanobacteria cells also aging from unicellular organisms to complex multicellular gregate together to provide the shading required for pro- organisms. Such multicellularity is an important mech- tection of photosynthetic machinery from damage by anism of yeast populations to retain resilience against excessive light [385]. Unlike yeast cells in a clump fluctuations in sucrose concentrations[376]. that usually stick together, the tightness of cyanobacte- Bistable response of cyanobacteria exposed to in- ria cell aggregation can increase or decrease as the light creasing light. Cyanobacteria are blue-green bacteria conditions change. Shown in Fig. 38, cells aggregating that are abundant in the environment. They are among together in blue light that is harmful to them, but re- the world’s most important oxygen producers and car- lax their clumps to optimize light capture in green light bon dioxide consumers [377]. For cyanobacteria, light [383]. Thus, the resilience of cyanobacteria populations is the main source of energy. Light is absorbed by could be enhanced by such regulations of cell-cell con- the phycobilisomes and photosystems is converted into tacts. chemical energy through photosynthesis [378]. The Bacterial antibiotic responses. The extensive use of color (wavelength) and intensity (irradiance) of light antibiotics has resulted in a situation where multidrug- could affect the growth of cyanobacteria. For example, resistant pathogens have become a severe menace to hu- the growth rates of the cyanobacteria were similar in or- man health worldwide [386]. Antibiotics affect bacte- ange and red light, but much lower in blue light [379]. rial cell physiology at many levels, and bacteria respond It is important to note that under a certain range, in- to antibiotics by changing their metabolism, gene ex- creasing the light intensity increases the growth rates of pression, and possibly even their mutation rate [387], cyanobacteria, yet strong light is harmful to cyanobac- which affects the level of their growth rate and ability teria. It may even cause the collapse of cyanobacteria to survive. Bacteria cell may survive an antibiotic treat- populations [380]. Understanding how cyanobacteria ment because individual cells may become tolerant or respond to light could improve photosynthetic efficiency adapt to the treatment, or because the population recov- and overall resilience. ers by recolonization, reproduction or rapid regrowth, The phenomenon that the rate of photosynthesis de- after some individual cells were killed [388, 389]. Such creases with increasing light is caused by photoinhi- mechanisms can be characterized by the resistance- bition. It manifest itself as a series of reactions that resilience framework in the field of microbiology [390], inhibit the activities of photosystem and it is apparent with resistance defined as the insensitivity to the treat- in cyanobacteria populations and phytoplankton species ment, and resilience as the time required for a commu- sensitive to high light [381, 382]. The existence of pho- nity to recover its former composition and functions af- toinhibition forces cyanobacteria to carefully balance ter a treatment [372]. The involvement of biological harvesting sufficient photons to maximally drive photo- mechanisms related to antibiotic resistance and antibi- synthesis while avoiding the damaging effects of excess otic resilience has been gaining concern in the scien- energy capture [383]. This balance is achieved by pho- tific community [390, 387, 391, 392, 372], contributing totaxis. Positive phototaxis is cell movement towards to the development of new treatment strategies to cope 36 Figure 36: Cooperative growth of yeast in sucrose leads to bistability and a fold bifurcation. (A to D) Individual populations starting at different initial densities were grown in 2% sucrose with daily dilutions into fresh media. Small populations below a critical density went extinct (red traces), whereas larger populations converged (blue traces) and maintained a stable density. (E) The stable and unstable fixed points measured by experiments are shown as symbols. Source: Figure from [254].

class of antibiotics. In the absence of antibiotic treatment, the population of ESBL-producing bacteria grows approximately exponentially until the growth rate decreases due to the limited nutrients, and ends up in the carrying capacity, as the blue/black curves shown in Fig. 39A. The time needed for a population to reach 50% of its carrying capacity is denoted as T 50%. If antibiotics are introduced, the time required for the bacterial population to reach its carrying capacity increases, and the higher antibiotic concentration is, the longer time is required to reach carrying capacity. as the green and yellow lines show in Fig. 39A. Antibiotic resilience is defined as the the inverse of the treated population’s T 50% 50% 50% (TA ), normalized by the untreated population’s T 50% (T0 ), which is written as T 50% Resilience = 0 . (3.3) 50% Figure 37: Critical transition in cyanobacteria population as the light TA intensity increases. Source: Figure from [384]. Antibiotic resistance is defined as the ratio between the minimum net growth rate in a treated population (ρA) and the net growth rate of an untreated population at the with and prevent the rise of resistant pathogenic bacteria same time (ρ0), which takes the form [387]. ρ Resistance = A . (3.4) ρ0 As shown in Fig. 39B, the higher antibiotic concentration is, less resistant the population becomes. Us- ing the resistance-resilience framework can visualize the shift in a population’s antibiotic response, and once the population undergo a crash, resistance is minimized Figure 38: Cell aggregation: shield from harmful light wavelengths. The increasing or decreasing the aggregation depends on the color of and resilience dominate the survival (Fig. 39C). This light. resistance-resilience framework effectively reveals the Source: Figure from [383]. phenotypic signature of each strain (Fig. 39D) when treated by a β-lactam. It reveals that an effective treat- Meredith et al. [371] apply the resistance-resilience ment should minimize both resistance and resilience. framework to the analysis of bacterial pathogens that In summary, the resilience in unicellular organisms produce extended-spectrum β-lactamases (ESBLs). ES- depends on the genotype and underlying molecular con- BLs are becoming increasingly prevalent and can de- nections within individual cells and the cooperations be- grade many β-lactam antibiotics—the most widely used tween cells. The latter can improve cell growth, which 37 Figure 39: Quantifying antibiotic resilience and antibiotic resistance. (A) When no antibiotic was added, the population transition nearly exponentially to a carrying capacity (black curve). When the antibiotic concentration increases, the time required to recover increases and the resilience decreases. (B) Net growth rate quantifies population’s resistance. (C) Resistance and resilience as functions of the cefotaxime concentration. (D) The resistance-resilience map defines a phenotypic signature. Source: Figure from [371]. in turn encourages more cooperation. This forms posi- discussed in Sec. 3.1.3 as an example, its four stable tive feedback and leads to bistability in the density of states could be mapped into four potential wells with unicellular populations. The multicellular organisms different heights in a landscape, and the landscape could can be viewed as evolutionary of benefits of clumps or visualize the transitions between them (Fig. 40). clustering of cells. Thus, the insight gained from unicellular organisms can be can be extended to multicellular organisms.

3.2.3. Potential landscapes of cellular processes in complex multicellular organisms The dynamics in biological networks are usually high-dimensional and nonlinear. They exhibit a large number of stable steady states (attractors) [393, 394]. Mapping the state of a biological system into a ball (or a point) in a high-dimensional state space, enables us to map a move to an attractor into a particular trajec- Figure 40: Quasi-potential landscape of a double negative feedback tory in this state space. Such movements can be then loop [321], which has been empirically found to have four stable well described using a quasi-potential landscape, which states occupied with different probabilities. is motivated by the famous epigenetic landscape first Source: Figure is from [397]. proposed by Waddington [395]. In a quasi-potential landscape, each point represents one state of the bio- Quasi-potential landscape has been a useful tool for logical network at equilibrium and the hight of its quasi- understanding the phase transitions within biological potential energy is inversely correlated to the probability systems, especially for the resilience of complex mul- that the system is found in that state [396]. An attractor ticellular organisms [397, 398, 399, 400]. In order to state is at the bottom of a valley (a potential well), and draw the quasi-potential landscape, we may need to find is more likely to be occupied than states in the higher quasi-potential functions of biological networks. Gen- areas (the potential hills) [397]. In the landscape, the erally, the dynamics of a biological network can be de- states of biological networks seek the way down to the scribed as a series of continuous differential equation potential well, representing the equilibrium of the sys- [393]: dx tem shifting to an attractor. The landscape reflects in- = F(x), (3.5) herent properties of the biological system, showing how dt T much ‘energy’ barrier needs to be overcome to enable where x(t) = (x1(t), x2(t),..., xN (t)) represents N the system to transition between any pairs of attractors system variables (e.g., gene expressions, biologi- [299]. Taking the double-negative feedback loop [321] cal molecular concentrations, etc.) and F(x) = 38 T (F1(x), F2(x),..., FN (x)) describes the ‘forces’ act- of the cell cycle. In addition, through global sensitiv- ing to change corresponding variables’ state. In closed ity analysis upon landscape and flux, the key elements equilibrium systems without significant exchange of en- for controlling the cell cycle speed are identified, which ergy, materials, and information with the outside envi- helps in designing effective strategy for drug discovery ronments, such as proteins folding [401, 402], the local for cancer. dynamics can be determined by the gradient of the interaction potential energy [393]: ∂U(x) Fi(x) = − , (3.6) ∂xi and the potential function U(x) can be directly inferred as: Xn Z U(x) = − Fi(x)dxi. (3.7) i=1 However, most biological systems are non-equilibrium open systems. Their dynamics can not be fully captured by the pure gradients (Eqs. 3.6 and 3.7). Fortunately, we can use the master equation [403] discussed in Sec. 3.2.1 to describe the time evolution of the probability of the system staying in each state. Next, we will review the studies on the reconstruction and analyses of the landscapes of cellular processes, such as cell cycle, cell differentiation, and disease progress in multicellular organisms. The Mexican hat landscapes of cell cycles. We have Figure 41: Diagram of the mammalian cell cycle network. The reviewed the simple genetic circuit in cell cycle, which network includes four major complexes formed by cyclins and could generate bistability in Sec. 3.1.3. However, the cyclin-dependent kinases (cyclin/CDks) – cyclin D/Cdk4-6, cyclin whole process of cell cycle is far more complex than E/Cdk2, cyclin A/Cdk2, and cyclin B/Cdk1 – which together determine the cell cycle dynamics. The mutual repression regulations what was presented there. The cell cycle is a series of between the tumor suppressor retinoblastoma protein (pRB) and the events that take place in a cell during its replication and trascription factor E2F control the cell cycle progression. division. It comprises several distinct phases: G1 phase Source: Figure is from [398]. (resting), S phase (synthesis), G2 phase (interphase), and M phase (mitosis) [398] shown in Fig. 42A. Activa- Another typical example is a well-studied mam- tion of each phase is dependent on the proper progres- malian cell cycle network [407, 408], whose detailed sion and completion of the previous one, which is mon- diagram is shown in Fig. 41. The network involves itored by cell cycle checkpoints. Plenty of studies has four major complexes formed by cyclins and cyclin- been carried out to uncovering the mechanisms under- dependent kinases (cyclin/CDks), centered on cyclin lying the cell cycle process, both in unicellular organ- D/Cdk4-6, cyclin E/Cdk2, cyclin A/Cdk2, and cyclin isms, such as budding yeast [404, 405], and multicellu- B/Cdk1. Together, they determine the cell cycle dynam- lar organisms, such as Xenopus laevis [406] and mam- ics [407, 398], which can be described by a set of 44 mals [398]. It has been found that the landscape and nonlinear ordinary differential equations. Li et al. [398] flux framework could be effectively used to explain the use the probability landscape and flux to determine the whole process of cell cycle and its checkpoints [404]. main driving force for the dynamics in the mammalian For example, Zhang et. al [406] quantify the underlying cell cycle network. The landscape reflects directly the landscape and flux of Xenopus laevis embryonic cell cy- steady state probability distribution Pss, which is deter- cle. The authors also uncover the corresponding Mex- mined by using a self-consistent mean field approxima- ican hat landscape with several local basins and barri- tion [400], and the potential landscape U (U = −lnPss), ers on the oscillation path was uncovered. The local giving the weight of each state. By projecting such basins characterize the different phases of the Xenopus 44-dimensional landscape to a two-dimensional state laevis embryonic cell cycle. The local barriers represent space, the landscape of the mammalian cell cycle sys- the checkpoints responsible for a global quantification tem in terms of two key proteins CycE and CycA is 39 Figure 42: Potential landscape of a complete cell cycle network. (A) The four phases of cell cycle with the three checkpoints: G1, S, G2, and one M phase. (B) Global landscape of cell cycle that contains three phases and two checkpoints. (C) The 2D landscape, where white arrows represent probabilistic fluxes, and red arrows represent the negative gradients of potential. Source: Figure is from [398]. shown in Fig. 42 B, which has a Mexican hat shape. In experiments, leading to predictions and potential anti- the landscape, the red colored region represents high po- cancer strategies. tential (low probability for the system to reach), and the Epigenetic landscapes for cell-fate induction. In cell- blue colored region along the Mexican hat ring with low fate induction, a cell progresses from an undifferentiated potential (high probability). The low potential blue re- state to one of a number of discrete, distinct, differen- gion forms a circle of oscillation trajectory, which guar- tiated cell fates. To describe such process, Wadding- antee the resilience of cell cycle oscillation dynamics. ton’s epigenetic landscape [395] is probably the most Along the cycle path, the progression of a cell cycle famous and most powerful metaphor, where cells are is determined by two driving forces: potential barriers represented by balls rolling downhill through a land- for deceleration and curl flux for acceleration. The curl scape of bifurcating valleys [299, 409, 410, 411, 321]. probability fluxes are shown as the white arrows in the The Waddington’s landscape starts from a single val- 2D landscape (Fig. 42C), and the negative gradients ley, representing the single undifferentiated steady state. of potential landscape are represented by red arrows. As time goes on, alternative valleys appear, representing The force from the negative gradient of potential attracts multiple differentiated states. In the landscape, each val- the cell cycle into the oscillation ring. The flux drives ley represents a possible cell fate and the ridges between the cell cycle oscillations along the ring path. Further- the valleys maintain the cell fate once it has been cho- more, along the ring with heterogeneous potential, there sen. In Waddington’s landscape, undifferentiated state are three basin of attractions corresponding to three cell is unstable, which triggers the differentiation process. It phases (G0/G1, S/G2 and M) and two barriers repre- is true for the stem cells during embryonic development. senting two checkpoints (G1 and G2 checkpoint). However, in the adult, the multipotent undifferentiated The potential landscape can help us to understand the cells are stable. Wang et al. [412] develop a framework role of attractors in the whole cell cycle dynamics quan- to quantify the Waddington landscape for a simple gene titatively. This landscape also provides a simple physi- regulatory circuit that governs binary cell fate decision cal explanation for the mechanism of cell-cycle check- module, through construction of underlying probability points. In addition, the influence of external or inter- landscape for cell development. The circuit consists of nal perturbations on the resilience of cell cycle could be two mutually inhibit transcription factors, which can be learned by doing global sensitivity analysis [398]. Such described by the minimal system equations [411]. In analysis quantifies the changes in the barrier heights, pe- such quantified landscape, the undifferentiated state can riod, and flux when parameters (regulation strengths or still be stable but has a small finite chance to climb up synthesis rates) are changed. By selecting the highly (induced by fluctuations) from the basin of attraction influential parameters in terms of the genes and reg- and escaping to the differentiated states (Fig. 43). ulations of the network, key elements or wirings that In Fig. 43, the depth of two valleys, which represent control the stability and the progression of cell cycle the potential distribution along the trajectories that start- are identified. The results can be also verified through ing from the same bifurcation point looks quite even. 40 network is transformed into a dynamic Bayesian network parameterized by the Monte Carlo Markov Chain method (Fig. 44B). This network is used to simulate the evolving stochastic characteristics. Since the exact transduction of extracellular signals to transcript factors in human embryonic stem cell is unknown, estimation of the landscape was needed, Chang et al. [276] choose to manipulate the expression levels of the three key regulators (Oct4, Sox2 and NANOG), mimicking the extracellular conditions. The joint probabilities of all the nodes in the network with Oct4/Sox2/NANOG set to various activity levels were calculated. The integration of these probabilities was used to estimate the network landscape. Shown in Fig. 44C, two states having significantly higher probabilities than the rest of the states. The states represent the human embryonic stem cell and its differentiated states. In the human embry- Figure 43: The quantified Waddington developmental landscape and onic stem cell state, all the embryonic stem markers are pathways Source: Figure is from [412]. active (1) and all the differentiation markers are inactive (0). While in the differentiated state, the activity compositions are opposite. These two states are separated by However, due to the complexity and noise (such as the barrier states with smaller probabilities, which prevents molecular noise discussed in Sec. 3.2.1), the landscape transitions between cell types by noise. for the cell developmental processes of multicellular or- In addition, through the global sensitivity analysis of ganisms maybe quite bumpy, and during the cell differ- parameters or connections between genes in the human entiation process, stochastic state transitions may hap- stem cell developmental network, Li et al. [409] quanti- pen [413, 414, 415]. For example, Wang at.al [400] con- tatively predict which connection links or nodes (genes) sidered the effect of stochastic fluctuations in a canon- are critical to cellular differentiation or reprogramming. ical gene circuit. They use the Fokker-Planck equation The results can be directly tested from the experiments. to reconstruct the potential landscape. They find that the The identified key links can be used to guide the differ- system can hop from one stable branch (e.g. pluripotent entiation designs or reprogramming tactics. state) to another (e.g. either one of two differentiated Quantifying the underlying landscape for cancer state) with some probability, even when it doesn’t reach networks. Cancer is believed to be a genetic disease bifurcation. The higher the noise level is, the higher arising from the accumulation of multiple genetic and probability for random state transitions. Fortunately, or- epigenetic alterations [416, 417]. It has long been rec- ganisms develop mechanisms to prevent this stochastic ognized as an evolutionary process [418], whose phys- effect [365]. Next, we show a specific example on cell ical mechanisms underlying cellular process of transi- differentiation. tions from normal to cancer states could be effectively Chang et al. [276] estimate the epigenetic landscape studied by using the idea of ecological resilience [419]. for a genetic network involved in regulating pluripo- Landscape analysis is a crucial tool for quantifing and tency and human embryonic stem cell differentiation. visualizing the state transitions between normal and This genetic network (Fig. 44A) was constructed by cancer states [397]. Next, we review the studies on un- starting with a set of marker genes of pluripotency and covering the landscape for cancer systems both at the differentiation lineages. Then, the authors collected the molecular and cellular levels. regulatory paths between any pair of genes. The re- Yu et al. [277] constructed a reliable gene regula- sulting network consists of direct regulatory interactions tory network for breast cancer. This network consists between 52 nodes, which includes three key regulators of 15 genes that are crucial for breast cancer, and 39 of embryonic stem cell (Oct4, NANOG and Sox2), six regulatory relations between them. As shown in Fig. protein complexes, and marker genes for the differenti- 45A, this gene regulatory network contains four onco- ation lineages. To calculate the probability of each cell genes (BRCA1, MDM2, RAS, HER2), three tumor sup- state, each protein in the network is considered as a bi- pressor genes (TP53, P21, RB), five kinases (CHEK1, nary variable, either active or inactive. Then the genetic CHEK2, AKT1, CDK2, RAF) being essential for the 41 Figure 44: Epigenetic landscape for the human embryonic stem cell network. (A). Genetic network regulating self-renewal and differentiation of hESC. (B). Bayesian Networks (2TBN) model of genetic network in hESC. (C). Illustration of the cell-state potential landscape. The color represents the potential of the cell state. The higher the potential is the a smaller probability of reaching that particular cell state. Source: Figure is from [276]. maintenance of cell cycle regulations, two genes (ATM, ATR) important for signal transduction, and a transcription factor (E2F1). The interactions between genes include both activation and repression regulations. The temporal evolution of the dynamics of this gene network is determined by the driving force involving gene regulations defined as follows

m1 n m2 n Figure 45: (A). The diagram for the gene regulatory network of dX X a j ∗ X j X b j ∗ S i = F = −K ∗X + + , (3.8) breast cancer, which contains 15 nodes (genes) and 39 edges (26 dt i i i S n + Xn S n + Xn j=1 j j=1 j activation interactions and 13 repression interactions). (B). The tristable landscape of the breast cancer gene regulatory network. Source: Figure is from [277]. where Xi represents the expression level of gene i (i = 1, 2, ..., 15), and three terms on the right side of the equation are self-degradation, activation and repression, respectively. Parameters K, a and b are constants; S rep- the rules they normally follow. The premalignant state resents the threshold of the sigmoid function; m1 (m2) is is a condition in which the cells grow with some abnor- the number of nodes that activate (repress) node i, and mal features resembling certain cancer characteristics. n is the Hill coefficient. Based on the 15 dynamic equa- In the cancer state, cell growth becomes uncontrollable tions and the self-consistent mean field approximation, and eventually spread to other organs of the body. The the steady-state probability distribution Pss is obtained progression of breast cancer can be seen as switching together with the potential landscape U = −lnPss. transitions between different state basins. The transi- Since it is difficult to visualize the landscape in a 15- tions between normal and premalignant states are al- dimensional space, Yu et al. [277] projected the land- most reversible, while those between premalignant and scape onto a 2-dimensional subspace spanned by the ex- cancer states are irreversible, which clearly illustrate the pression levels of BRCA1 (an oncogene of breast can- mechanisms of cancerization [417]. In addition, global cer) and E2F1(a biomarker of breast cancer). sensitivity analysis shows that changing the strengths of As shown in Fig. 45B, there are three attractor basins the key regulations in the breast cancer gene regulatory on the phenotypic landscape, representing the normal, network can allow the landscape topography to move premalignant and cancer state, respectively. In the nor- in preferred directions that are beneficial for reversing mal state, the cell growth, arresting and apoptosis obey cancer back to normal state. 42 Besides uncovering the epigenetic landscape from stable state (characterized by the distance between the gene regulatory network (molecular layer) of cancer stable and unstable fixed points) shrinks, elevating the [399, 420, 277], the cancer–immune interaction net- chance of extinction by stochastic perturbations. Dai et work (cellular layer) is also important for understanding al. [254] test the resilience of yeast populations at dif- tumorigenesis and the development of cancer and im- ferent dilution factors by salt shock of sodium chloride pact of immunotherapy [421, 422]. Li et al. [422] con- for 1 day, and find that populations at low dilution fac- struct a cancer immune network consisting of 13 cells (a tors are able to recover from the perturbation, whereas cancer cell and 12 types of immune cells) and 13 related those at high dilution factors go extinct. factors cytokines, and uncover the underlying mecha- When a biological population approaches a tipping nism of cancer immunity based on landscape topogra- point, recovery rates from small perturbations are ob- phy. In the landscape, three steady states (normal, low served tend to zero [380]. Thus, in the vicinity of bi- cancer, and high cancer states) appear. Upon certain furcation in yeast populations [254], the critical slow- cell-cell interactions, limit cycle oscillations emerge, ing down phenomenon is directly observed. Since in which are common in the immune system [423], pro- this vicinity, the populations become more vulnerable viding a physical view of tumorigenesis and cancer re- to disturbance and more time is needed to recover from covery processes. small perturbations, the system becomes more corre- In summary, the landscape and flux theory has been a lated with its past, leading to an increase in autocor- powerful tool for quantifing and visualizing the dynam- relation between density fluctuations at different time ical transitions between different stable states in various points. In addition, among more than 46 replicate yeast important biological processes in multicellular organ- populations, the size of fluctuations in population den- isms, such as cell cycle [424, 406], cell differentiation sities for five days increases as the dilution factor in- [410, 425], the initiation and development of disease creases, so that the standard deviation and coefficient [426, 417]. Based on the epigenetic landscape, key el- of variation both increase [254]. These three indicators ements and links are identified through global sensitiv- based on critical slowing down are found to be good ity analysis, which quantifies how much perturbations a early warning signals. However, skewness, a suggested steady stable states can stand. early warning signal not based on critical slowing down, which measures the asymmetry of fluctuations in pop- 3.3. Indicators of resilience in biological systems ulation density [429], is not a good warning signal for yeast populations collapse [254]. Thanks to the presence of bistability (or multistabil- The three indicators (autocorrelation, the standard de- ity), a biological system can shift abruptly from one viation, and coefficient of variation) based on critical state to an alternative stable state at a tipping point slowing down discussed above are measured from the [380], Once the shift happens, reversing it could be dif- time series of population density, which demands obser- ficult [427]. Thus, finding early warning signals before vations over a long time span. Dai et al. [30] identify in- the catastrophic shift has important implications for pre- dicators based on spatial structure as early warning sig- venting biological population collapses by effective hu- nals for yeast population collapse. They spatially extend man interventions, and the onset of diseases. Next, we the yeast populations [254], and the spatial coupling review studies on detecting effective indicators as early between local populations is introduced by transferring warning signals before biological population collapse 25% of a local population to each of its nearest neigh- and the onset of diseases. bours. Consistent with critical slowing down, clear increases in the coefficient of variation and autocorrela- 3.3.1. Early warning signals before biological population of connected populations towards the tipping point tions collapse are observed. However, the magnitudes of increase in As discussed in Sec. 3.2.2, due to the cooperative coefficient of variation and autocorrelation are smaller growth of budding yeast in sucrose, bistability and a than those in the isolated populations (Fig. 46). Such fold bifurcation appear when the dilution factor changes suppression of the two leading indicators in connected [254, 428]. As shown in Fig. 36, the fold bifurcation oc- populations is caused by the averaging effect of disper- curs when the stable and unstable fixed points “collide”. sal. In an extreme case, where ten populations are mixed For higher dilution factors, populations always collapse, completely each day, the populations show almost no as extinction is the only stable state. For lower dilution increase in variation before the tipping point. However, factors side, near the bifurcation, a population becomes the spatial coupling introduces another warning indica- less resilient because the basin of attraction around the tor: recovery length, which is the distance necessary 43 for connected populations to recover from spatial per- fitness-related phenotypic traits [431]. For instance, turbations. The recovery length increased substantially declines in the mean body size of stressed popula- before population collapse, suggesting that the spatial tions occur significantly prior to their collapse or ex- scale of recovery can provide a superior warning signal tinction [432]. When cyanobacteria population ap- before tipping points in spatially extended systems [30]. proaches a tipping point, the photosystem II quantum In addition, Rindi et al. [430] observe that the recovery yield (a chlorophyll concentrations association indica- length increases when the marine benthic system ap- tor) decreases significantly [380]. Berghof et al. [433] proaches the tipping point, where the system shifts from show that there is genetic variation in resilience indi- a canopy-dominated to a turf-dominated state, provid- cators based on body weight deviations in layer chick- ing field-based evidence of spatial signatures of critical ens. Under fishing stress, the maturation schedules in slowing down in natural conditions. cod populations significantly change before the population collapse [434, 435]. In addition, recent studies show that combined together trait-based and traditional density-based indicators can not only provided significantly more reliable early warning signals but also generate reliable signals earlier than using abundance data alone [432, 436, 431]. For instance, the mean or variance in body size combined with fluctuations in population density could show a greater indication of critical transition in protist populations than each of the two indicators alone [432]. Thus, before biological populations collapse, statistical indicators based on critical slowing down both in temporal and spatial fluctuations in population density, skewness and phenotypic trait-variation are efficient early warning signals. On one side, as discussed above, not all the indicators always perform well in certain population collapses. For example, skewness is not a good indicator for yeast population collapse [254], and the variance in population density fluctuation does not increase before the collapse of cyanobacteria population when the light intensity increases [380]. On the other side, some of them may appear at the same time for the same system. For example, Drake et al [199] conduct an experiment with replicate laboratory populations of Daphnia magna, which are induced by the controlled decline in environmental conditions. Four statistical indicators, i.e., coefficient of variation, autocorrelation, skewness, and spatial correlation in population size all show evidence of the approaching bifurcation as early as 110 days ( 8 generations) before the transition Figure 46: Early warning signals based on fluctuations show occurring. suppressed increase in connected populations. The coefficient of variation (A) and temporal correlation (B) of both isolated 3.3.2. Critical slowing down as early warning signals populations (red squares) and connected populations (blue circles) for the onset of disease increased before the tipping point. The signals were suppressed in the connected populations, possibly owing to the averaging effect of In humans and animals, resilience is the capacity to dispersal. be minimally affected by disturbances or to rapidly re- Source: Figure is from [277]. turn to the state maintained before exposure to a disturbance. Less resilient people or animals are expected Besides the signals based on critical slowing down, to be more susceptible to environmental perturbations, trait variation has been identified as early warning sig- which may lead to diseases or deaths [437, 433]. Due to nals for population collapse, especially the change in the inherent complexity and nonlinear dynamics, tran- 44 sitions from health to disease are usually not continu- acterized by a wide array of symptoms, such as inability ous but revealed by sudden shifts in system states [438]. to sleep, low mood, loss of interest in previously en- Treating a person or an animal as a dynamical sys- joyed activities, and suicidal tendencies [438]. These tem, ‘healthy’ and ‘diseased’ are two alternative stable symptoms are correlate together, forming a network of states, which can be modeled as two basins in the sta- symptoms. For example, a person may become de- bility landscape (Fig. 47A and B). When the dynamics pressed through the following causal chain of feelings of an individual has a high resilience and is far away and experiences: stress → negative emotions → sleep from the tipping point, it is difficult to move the ball problems → anhedonia [449, 450]. In the network of (representing the state of the individual) from ‘healthy’ symptom, positive feedback loops exist, such as worry- into ‘diseased’, since the basin representing ‘healthy’ ing → feeling down → even more worrying or feeling state is deep and wide. As the dynamics of an individ- down → engaging less in social life → feel even more ual approaches a tipping point, the ‘healthy’ basin be- down [451]. Such positive feedback loops can cause a comes shallow and narrow, increasing chances for tran- system to have alternative stable states, and gradually sition from ‘healthy’ to ‘diseased’ by stochastic fluctu- changing external conditions may cause the system to ations. Once the catastrophic shift occurs, it is difficult approach a tipping point. Thus, the onset and remis- to reverse. That is why early diagnosis is important in sion of clinical depression may occur suddenly. Leem- medicine and a trend from diagnosing disease to pre- put et al. [452] show for a large group of healthy indi- dicting disease emerges [439], especially for chronic viduals and patients that the probability of an upcom- diseases. An increasing evidence shows that critical ing shift between a depressed and a normal state is re- slowing down could be a source of efficient early warn- lated to statistical indicators of critical slowing down. ing signals for the onset of diseases. Next, we review The authors monitor the time series of emotion scores examples of detecting the early warning signals before for four observed variables, representing four emotions: the onset of diseases. cheerful, content, sad and anxious. Among the gen- Epilepsy is a central nervous system disorder eral population sample, 13.5% subjects show transitions in which brain activity becomes abnormal, causing from normal states to clinical depressed states. In indi- seizures or periods of unusual behavior, sensations, and viduals who are close to a tipping point, both temporal sometimes loss of awareness [441]. A national survey of autocorrelation and variance of fluctuations in emotion the UK population reasserts that epilepsy and seizures scores are higher than in individuals that are far away can develop in any person at any age [442]. Epilepsy from emotional transitions (Fig. 47). This difference makes living a normal life difficult for patients because suggests that critical slowing down could be an early seizures start and terminate suddenly, are not easy to warning signal for onset and termination of depression. predict [438, 443]. Since 1970s, researchers have been In addition, Wichers et. al [453] directly observe ris- seeking ways to predict the occurrence of seizures from ing early warning signals pattern in individual’s critical the electroencephalogram (EEG) of epilepsy patients transitions in depressive symptoms, based on long-term [444]. For example, Meisel et. al [445] find that a (10 times a day over 239 days) emotion monitoring data. Hopf bifurcation could be involved near seizure onset, The ability to anticipate transitions between healthy and and increased variance in spiking patterns of individual diseased states could prove beneficial in terms of tim- neurons has been proposed as an early warning signal to ing and magnitude of treatment interventions, which is detect the onset of a sudden epilepsy seizure. However, essential in optimizing health care. most of the previous studies on prediction that yielded Besides the applications in epilepsy and clinical de- rather promising results were recently found to be ir- pression, early warning signals, especially those based reproducible [446]. For instance, Wilkat et al. [447] on critical slowing down, have been found in the onset investigate long-term, multichannel recordings of brain of other diseases. For example, Quail et al [454] de- dynamics from 28 subjects with epilepsy, and find no tect early warning signals to predict the onset of abnor- clear-cut evidence for critical slowing down prior to 105 mal alternating cardiac rhythms. They treat embryonic epileptic seizures. Whereas, for the critical transitions chick cardiac cells with a potassium channel blocker, from the ictal and post-ictal states of a seizure (the self- which leads to the initiation of alternating rhythms, and termination of seizure), human brain electrical activity associate the transition with a period-doubling bifurca- at various spatial scales exhibits the common dynamical tion. When the system approaches the bifurcation, its signature of an impending critical transition indicating dynamics slow down, and noise amplification and oscil- the critical slowing down [448]. lations in the autocorrelation function appear in the ag- Clinical depression is a serious mood disorder, char- gregate’s interbeat intervals. Based on the return maps 45 Figure 47: Critical slowing down is a generic indicator that a patient has lost resilience in the sense that the patient may shift more easily from his current ‘healthy’ state into an alternative ‘diseased’ state. Comparing to the high resilience cases (A, C, D and G) that are far away from the tipping point, if the patient is close to the tipping point (B), three statistical indicators: recovery time (E), variance (F), and autocorrelation (H) for critical slowing down all increase. Source: Figure is from [440]. that relate the current interbeat interval with the follow- above, are based on the specific phenotypic data. Ac- ing interbeat interval, its slope can be used to assess how cording to the central dogma, inside cells of humans far the system is from the transition. Hsieh et al. [455] and animals, gene expression is the cornerstone for all develop a statistical indicator based on probabilistic risk the cellular activities. In addition, thanks to the devel- assessment framework to predict and access the ozone- opment of high throughput technologies, massive gene associated lung function decrement, and propose a com- expression data has been accumulated [457, 439]. It is posite indicator as predictor, which includes standard crucial and necessary for making it feasible to evalu- deviation, coefficient of variation, skewness, autocorre- ate effective early warning signals for critical transitions lation and coefficient of spatial correlation. Tambuyzer in biological processes, especially the development of et al. [456] find that amplitude increases of interleukin- diseases, based on gene expression profiles. In 2012, 6 fluctuations of individual pigs can be used as indicator Chen et al. [458] theoretically derive an index based of the infection state. on a dynamical network biomarker (DNB) that serves In addition, for noisy biological systems, the criti- as a general early-warning signal indicating an immi- cal slowing down do not always indicate an upcoming nent bifurcation or sudden deterioration before the criti- critical transition. Due to the decreased stability of the cal transition occurs. The authors validate the relevance attractor, systems may exhibit flickering between two of DNB and diseases by related experimental data and states until the alternative attractor is eventually gaining functional analysis. Next, we review studies on identi- stability and becoming the new stable state [439], which fying DNB based on time-series gene expression data, has been found in the onset of paroxysmal atrial fibril- to predict critical transitions in biological processes. lation [440] and epilepsy [448]. In conclusion, the development and experimental validation of early warning Given the gene expression profiles of a number of signals for the onset of diseases is a promising direction, genes from several samples or across different experi- which can be used for future therapeutic applications, mental conditions, the correlations between gene’s ex- such as prediction of therapeutic responses and clinical pression levels can be calculated, forming a gene co- outcomes, and for the design of personalized treatments. expression network [459]. During the development of a complex disease, the expression levels of certain genes 3.3.3. Dynamic network biomarkers in the progression could change significantly, such as oncogenes are over- of complex diseases based on gene expression expressed [460] and tumor suppressor gene are under- data expressed [461] in many cancers. Thus the correlations The studies on detection of early warning signals between genes will also change, leading to change in of critical transitions in biological systems discussed network’s structure. By analyzing the nonlinear dynam- 46 ics in gene expression near the bifurcation point, Chen reverse [438]. In order to detect the pre-disease stage et al. [458] prove that there exists a group of genes (or of type 2 diabetes, Li et. al [466] apply the DNB-based more generally molecules), called DNBs, with the Pear- method [458] discussed above to the temporal-spatial son correlation coefficients between DNBs (PCCd) in- gene expression data of rats in different stages of type creasing while the correlations between DNBs and non- 2 diabetes, and find that there are two different critical DNBs (PCCo) decreasing, when the system reaches states during type 2 diabetes development, characterized the pre-disease state. In addition, the average standard as responses to insulin resistance and serious inflamma- deviations of DNBs (SDd) drastically increase, which tion. They also show that most of DNB genes, in par- coincides with the critical slowing down phenomenon ticular the core ones, tend to be located at the upstream [452]. As shown in Fig. 48, the connections between of biological pathways. This indicates that DNB genes DNBs become intense and the standard deviations of act as the causal factors rather than the consequence of the DNBs’ states increase, when the gene co-expression driving the downstream molecules to change their tran- network is in the pre-disease state. Chen et al. [458] scriptional activities. For type 1 diabetes, two DNBs further propose a composite index I = SDd·PCCd , which are obtained as early warning signals for two critical PCCo increases significantly in the pre-disease state and have transitions leading to peri-insulitis and hyperglycemia been verified in complex diseases, such as the acute lung in non-obese diabetic mice [467]. Moreover, Zeng et injury and liver cancer [462]. al. [468] identify the modules present at the pre-disease The implementation of such framework to detect stage based on dynamical network biomarkers, which DNBs [458] requires multiple case samples in each time can serve as warning signals for the pre-disease state. point to calculate reliable Pearson correlation coeffi- Influenza is an infectious disease caused by influenza cients, which limits the framework application to indi- virus and it spreads around the world, leading to global vidual pre-disease state prediction. Due to the strongly respiratory illness and even deaths. The development fluctuating and correlated nature of DNBs in the pre- of single-sample based DNB method makes it possi- disease stage, their normalized expression levels have ble to predict in advance influenza at individual level. a double-peak distribution, but the non-DNBs have a Based on the temporal gene microarray data of hu- single-peak distribution in the pre-disease stage. Thus man influenza infection caused by H3N2 virus, Liu et. Liu et al. [284] use Kullback–Leibler divergence, which al [284, 463] successfully identify pre-disease samples measures the difference between two data distributions, from individuals before the emergence of the serious to formulate the DNB single-sample score to identify disease symptoms. In addition, influenza has the char- the pre-disease state of a disease on the basis of a sin- acteristic of seasonal collective outbreaks. In order to gle case sample. This facilitates early diagnosis before find the early warning signals preceding the influenza the disease state or the occurrence of serious deterio- outbreak, Chen et. al [469] collect the historical longi- ration. In addition, near the bifurcation point, DNB tudinal records of flu caused hospitalization from 278 biomolecules exhibit significantly collective behaviors clinics distributed in 23 wards in Tokyo, and 225 clin- with fluctuations, which results in the local entropy in- ics distributed in 30 districts in Hokkaido, Japan, from creasing. Therefore, an index called single-sample land- 2009 to 2016. They construct a ward-network based on scape entropy score (SLE score) can be used to predict the actual locations of wards and their adjacency rela- critical transition of biological processes [463]. Next, tionships. By applying a local network DNB-based in- we show the applications of such DNB-based methods dex to predict influenza outbreaks in each ward, Chen to detect early warning signals for the critical transitions et. al [469] detect early warning signal with an average in the development of diseases and other biological pro- of 4 week window lead prior to each seasonal outbreak cesses. of influenza. Signaling the onset and deterioration of diseases. Acquired drug resistance in cancer cells is considered Diabetes is one of the most common chronic diseases as the major reason why patients fail to respond to can- and has two major subtypes: type 1 diabetes caused by cer therapies. Generally, acquired drug resistance can the failure of pancreas to produce enough insulin due be regarded as a process of biological network evolu- to the loss of beta cells [464]; and type 2 diabetes re- tion so as to make the system adapt to the drug envi- sulting from insulin resistance, when blood sugar can- ronment [470]. According to the biological feature of not enter cells [465]. The progress of both subtypes the time-dependent progression of MCF-7 breast cancer contains multiple stages (i.e. health stage, pre-disease cells exposed to tamoxifen, the process to acquire drug stage and disease stage), and the clinical diagnosis is resistance is divided into three stages: a non-resistance, usually made in the disease stage, which is difficult to a pre-resistance (or the tipping point), and a resistance 47 Figure 48: DNBs as early warning signals for two complex diseases. (A) and (C) show the expression profile of detected DNBs (red rectangle) and other randomly selected non-DNBs from the acute lung injury and the HBV induced liver cancer data set respectively. (B) and (D) are the corresponding gene co-expression networks, where red nodes represent the DNBs and their nearest neighbors are shown as grey nodes. For the acute lung injury, the dynamic evolution of the network structure for the identified DNB subnetwork is shown (E) and the whole mouse network including DNBs (F) is visualized, with the time point 8h being the critical point. Source: Figure is from [458]. state. Through the DNB approach, DNB network alter- view. In the single-cell level, cell fate induction requires ation is found to occur prior to the observation of tamox- broad changes of their gene expression profile, and cell- ifen resistance, and follows the appearance of mutation to-cell gene expression stochasticity could play a key genes [470]. Furthermore, distant metastasis of cancer role in differentiation. Mojtahedi et. al [473] show that cells is the main cause of cancer deaths. Detection of commitment of blood progenitor cells to the erythroid the tipping point before metastasis of cancer is critical or myeloid lineage is preceded by the destabilization of to prevent further irreversible deterioration. their high-dimensional attractor state. This causes dif- DNB-based methods have been applied to detect ferentiating cells to undergo a critical state transition. early warning signals for lung cancer [463] and other At the point of fate commitment, a peak in gene ex- cancer metastasis [471]. Recently, Yang et al. [472] pression variability appears [474]. Thus the structure analyse time-series gene expression data in spontaneous of gene co-expression network changes greatly. By us- pulmonary metastasis mice HCCLM3-RFP model with ing the DNB-method, a group of DNB genes for which DNB-based method, and identify CALML3 as a core the fluctuations and correlations between them increase DNB member, which has been further verified as a have been identified, which can be used as early warning suppressor of metastasis, thus providing a prognostic signals for the differentiation in primary chicken ery- biomarker and therapy target in hepatocellular carci- throcytic progenitor cells [475]. In addition, the criti- noma. cal differentiation state of MCF-7 cells for breast cancer can also be identified by using DNB method. This ap- Identitying the tipping point in other biological proach provides an opportunity to interrupt and prevent processes. The DNB-based method can not only be the continuing costly cycle of managing breast cancer used to predict the critical transitions in the develop- and its complications. ment of diseases, but also the critical transitions in other complex biological processes from time series gene ex- Immunotherapy using antibodies that block immune pression data. The cell-differentiation process has been checkpoints is an emerging success story for some seen as a stereotyped program leading from one progen- patients with cancer. However, majority of patients itor toward a functional cell, in the cell population-based gain no benefit but experience considerable toxicity 48 [476]. Plenty of efforts have been poured into identi- 4. Behavior transitions in animal and human net- fying biomarkers for predicting the response to immune works checkpoint blockade in cancer [476, 477, 478]. How- ever, so far no biomarker reliably predicts response in Similar to the resilience in ecological and biological a sufficiently rigorous manner for routine use. Fortu- networks discussed in the previous two chapters, the re- nately, the therapeutic response to immune checkpoint silience of a social network, defined as its ability to cope blockade is a critical state transition of a complex sys- with perturbations, shocks, or stresses, can take on dif- tem. The DNB-based methods have shown their abili- ferent forms in various settings. The resilience require- ties to predict the transitions from a pre-disease state to a ments can range from preservation of the entire system, disease state [463]. Thus, the DNB-based methods may or, specifically, the sustainability of its structure and op- be a potential tool for predicting immune checkpoint erational ability. We consider resilience in the following blockade. Lesterhuis et. al [476] propose that these three broad settings. dynamic biomarkers could prove to be useful in distin- In the first one, we consider humans social systems, guishing responses by non-responding patients to save which belong to the subject of traditional sociology. In them immunotherapy, as well as facilitate the identifica- this setting, we acknowledge two different notions of tion of new therapeutic targets for combination therapy. resilience. One focuses on something that we call cultural resistance of a society, which intuitively speaking requires that society evolves their opinions and beliefs In a sentence, DNB is a group of molecules that have in a smooth and orderly manner. The other type of resis- strongly correlated activities and fluctuations [472], tance, termed here survival resilience, is concerned with which can be used as early warning signals for the onset the very preservation of a social system. Intuitively, this and deterioration of complex diseases, as well as other kind of resistance requires that a society, or community, biological processes, such as cell fate induction, provid- can adjust itself in response to the external stresses or ing new ways to uncover underlying mechanisms be- challenges without self-destruction. hind various biological processes, and can benefit the The second considers resilience of social animals scheduling of treatments for complex diseases. and their communities. In this setting, we examine only survival resilience, which focuses on species and their communities preservation against environmental In the summary of all the studies on the resilience change and competition from other species and commu- in biological networks, we conclude that bistability, nities. Finally, the third setting recognizes the resilience or multistability, universally exists in biological net- of a system composed of two strongly linked subsys- works at different levels, ranging from genetic circuits at tems, social networks and ecological systems. Humans molecular level through unicellular populations to disas species have been so successful in spreading and ease networks at phenotypic level. They are generated dominating all ecosystems on the Earth that further un- by the underlying feedback loops and ultrasensitivity restricted growth of human population and increased ex- [297, 280] or particular network topology [294, 301]. ploitation of living and mineral resources of the Earth Due to the existence of alternative stable states, bio- may threaten the stability of a global ecosystem and logical networks could response to external or internal preservation of the human civilization. We will refer to stimulus in digital way. Once the stimulus exceeds a this complex system as the planetary social-ecological tipping point, critical transitions appear and the biolog- system to underscore its global range. The goal is to in- ical network shifts from one state to another abruptly, crease survival resilience of this complex system in the which is difficult to predict. But we could identify in- face of the increasingly limited resources of our planet dicators of resilience as effective early warning signals and the rapid change of the environment in response to for critical transitions in biological systems. They in- increasing impact of human activities on the Earth. Next clude coefficient of variance, autocorrelation based on we will introduce the two terms cultural resistance and critical slowing down and dynamic network biomark- survival resilience in social networks, focusing on hu- ers. The studies on the resilience in biological networks man social networks below. could help us understand the complex biological system Cultural resilience in social networks. Sociology has and design effective therapeutic methods, as well as find become a pioneer and early adopter of social network more applications in health management. Next, we will analysis. The first example known to us is an analysis review literatures on the resilience of social-ecological of a friendship network in a school class of year 1880-81 systems and social systems. that was made using mixed-methods by the German pri- 49 mary school teacher Johannes Delitsch [479]. A more continuity, and thus also of cultural resilience. Hence, advanced approach, similar to modern social network studies that focus on the discovery of conditions under analysis and called sociometry [480], is discussed in which shifts arise in social system opinions and beliefs Ref. [481]. This approach was used to analyze inter- are of great importance. Equally important is an ability actions of the inmates in a prison [482]. However, the to derive from system parameters the values of critical significant barrier to overcome was the issues of data points at which such shifts appear. Identifying measur- collection, which was laborious, imprecise, and prone to able early warning signals, or at least the distance to the misinterpretations of results. The breakthrough came in tipping point, is vital to the resilience policy develop- the 1990s from internet and wireless-based interactions. ment and the disaster avoidance strategies. They provided easily collected datasets scalable to mil- In the Naming Game studies, the critical tipping lions and even billions (e.g., Facebook) of members. points arise in the presence of the committed agents The network analysis has become a gold mine for social (also referred to as zealots) introduced first in Ref. scientists. It opened also network analysis to statisti- [500]. These agents never change the opinion that they cal physics, including novel applications of classic Ising hold initially and which they promote whenever they as- model [483, 484, 485] and new network models [2]. Al- sume the role of a speaker. The critical point value is a ternative approaches developed in a new branch of com- function of the fractions of the population belonging to putational sociology use agent-based computer simula- each of the committed minorities and the total number tions [486, 487]. All these development revolutionized of such communities. The small number of communi- the sociology [488]. ties with one of them sufficiently dominant in size guar- Social analyses demonstrate that the network struc- antees that the social system will rapidly reach a con- ture can enhance or weaken the network’s survival re- sensus state[500] on the opinion of the dominant com- silience. For instance, communities in a social net- mitted minority [497, 501]. On the over hand, a large work play a vital role in its resilience [489, 490]. Their number of small committed communities also guaran- strengths and dynamics are closely related to the amount tees consensus state, but the opinion of the consensus of social capital accumulated by the community mem- is independent of the initial size of the corresponding bers [491, 492, 493, 494]. In turn, this strength is essen- committed minority [502], creating potentially messy or tial for network resilience in response to a crisis or dis- even disruptive transition. Other outcomes are also pos- aster [495]. In general, the more significant is the social sible [501, 503]. Committed agents can also be defined capital accumulated within a community, the stronger in the voter model [504]. is the overall resilience of such networks compared to Inspired by the binary Naming Game model, a three- a homogeneous system with weak communities or no state model of social balance with an external deradi- communities at all. calization field is studied in Ref. [505]. The mean-field In the context of social systems, cultural resilience analysis shows the existence of a critical value of the ex- focuses on avoiding a drastic and disruptive change of ternal field strength. This value separates a weak exter- prevailing opinions and beliefs held by the social sys- nal field under which the system exhibits a metastable tem members. This kind of resilience is studied using fixed point from the strong external field under which models of social interactions, which enable the opin- there is only one stable fixed point. At the critical value, ions to evolve and innovation to spread through the in- the field is at a saddle point. This dynamic is similar fluence and persuasion exerted by personal interactions. to the dynamics of the Naming Game, proving that an Thus, these models do not consider changes imposed by external field influences the entire network at the same the sheer military or police force to subjugate the soci- time does not change dynamics significantly. eties. The models that represent these processes from Survival resilience. Besides the population level as the perspective of individuals include the voter model studied in cultural resilience, social resilience is also [496], the Naming Game (NG) [497], and the Threshold studied at the level of individuals, which is the lowest Model (TM) [498], and their variants. Additional mod- level of social systems. Such studies focus on how the els focusing on interactions of group behavior, such as outer environment could trigger fluctuations in the life flock/swarm behavior [499] are also applicable to simu- of an individual. This kind of research has stronger and lating social network dynamics. deeper roots in psychology, especially in developmental Whereas orderly and evolutionary changes are often psychology rather than in sociology or ecology. Pos- necessary to enable social systems to adapt to the evolv- itive psychology, including family-school partnership, ing environment, such changes might become so rapid or community support, have been observed to foster the and disruptive that they may result in loss of cultural resilience of individuals [506, 507]. More recently, the 50 role of social capital in communities to which individu- formal descriptions of social networks and their rela- als belong have been studied, often in the context of an tionships. Such descriptions originated in traditional individual’s reactions to traumatic events [508]. A nat- sociology but recently the sociologists also embraced ural extension of such studies is the research on com- socio-physics models, most of those, however, are of munity resilience’s impact on health and human devel- unproven validity. opment. Hence, it can be seen as an example of co- development of the resilience theory in the context of 4.1. Cultural resilience of social systems socio-ecological systems (SES’s) [508, 509, 510]. An- other example of such co-development focuses on urban Traditional approaches to social systems use the resilience of cities in response to the disaster, terrorism, macroscopic scale at which all inter-human interactions or other disturbances [511, 512, 513, 514, 515], which can be reduced to a set of mutual standards and pat- concentrates on one aspect of survival resilience. terns characteristic for interactions within groups and A study of the structural root causes of the resilience institutions of an underlying social system. Relatively of consensus in dynamic collective behaviors is pre- stable and frequently arising group types are often ele- sented in Ref. [499]. The authors construct the dynamic vated to a formal or legal organizational status such as signaling network to analyze the controllability of the marriages, families, corporations, or religious groups. group dynamics. The system is a small-world network. An example of such an approach is presented in Ref. The resilience (i.e., the alignment of opinions) is studied [518]. This reference introduces an AGIL scheme that in the presence of exogenous environmental noise. The distinguishes four core functions that collectively under- authors found that resilience strongly depends on the lie the stability and survival of a social system. They out-degrees of the nodes. The group exhibits a higher are as follows. Function (A) denotes adaptation since a level of resilience with larger out-degrees. Besides, system must adapt to its physical and social environ- when a single giant strongly connected component sur- ment as well as gradually accommodate the environ- vives the disturbance, it is organized into a large-scale ment to its needs. Function (G) denotes goal attainment coherent alignment of individuals. because a system must define and achieve its primary Social contagion theory describes peer effects, inter- goals. Function (I) stands for integration to enable a personal influence, and other events or conditions that system to coordinate and regulate the interrelationships could cause the emergence of crowd behavior. Based of its components and strive toward a cohesive whole. on an analysis of empirical datasets [516], the authors Finally, function (L) denotes latency so a system can describe the regularities that they discovered. Those furnish, maintain, and renew itself and motivate its in- discoveries motivated the authors to propose that hu- dividuals to perform their roles according to social and man social networks may exhibit a “three degrees of cultural expectations. At such a high level of generality, influence” property. They also review statistical ap- social resilience is defined by Parsons’ general theory proaches they have used to characterize interpersonal in which intra- and inter- systemic relations are charac- influence with respect to such diverse phenomena as terized by cohesion, consensus, and order imposed by obesity, smoking, cooperation, and happiness. the mentioned above four core functions. The aim is to The community’s resilience against disastrous events, represent the current status of social norms and rules to such as violent acts of terrorism, is studied in Ref. provide framework for analysis of the dynamics of the [517]. The author observes that the community is the lo- system. cus of response to disaster thanks to the amounts of “so- The opposite approach focuses on understanding re- cial capital” accumulated through interactions between silience from interactions of individuals endowed with members. The author analyzes the ways of enhancing roles and attributes, which enables analysis of human the capabilities that the community already possesses interaction dynamics and motivates the development of for dealing with disasters. One example is that the poli- the social network approaches. As discussed in the pre- cies addressing the threat of terrorism can be used to vious section, the cultural resilience in social systems increase the resilience of communities facing such a focuses on the continuous and orderly evolution of be- threat. liefs in order to avoid disruptions and discontinuities in There is a unique challenge of studying the resilience the culture. The prevailing approach in studying the dy- of social systems, when compared to natural sciences, namics of consensus formation [500] in social systems in which systems and interactions of their elements are relies on modeling elementary interactions between in- unambiguously and often formally defined. In contrast, dividuals and observing what stable or semistable con- this challenge arises because social sciences rely on in- sensus emerges from the simple elementary interaction 51 rules and the emerging social network structure. Ac- cordingly, we review the results on cultural resilience by discussing the approaches based on the most popular agent interaction rules.

4.1.1. Naming Game model The Naming Game model has been initially introduced to account for the emergence of a shared vocabulary created through social/cultural learning [519]. However, over time, the Naming Game has become an archetype for linguistic evolution and mathematical social and behavioral analysis [502]. Figure 49 shows an example of the Naming Game interaction rules. The Figure 49: The Naming Game interaction rule example. beautiful property of Naming Game is that it is a min- Source: Figure from [519]. imal model employing local communications that can capture generic and essential features of an agreement process in networked agent-based systems. Examples sensus can be reached [520]. They also analyze how are a group of robots, the emergence of shared commu- well the average convergence times and their distribu- nication schemes, or the development of a shared key tion describe the convergence of individual processes, for encrypted communications. identify the word that is going to dominate in the final In the Naming Game, agents perform pairwise inter- convergence stage, and, in case of the binary Naming actions to reach an agreement on the name to assign to Game, define the conditions for symmetry breaking. a single object. The interactions are limited to pairs of The study of the dynamics of the original Naming neighboring nodes, as defined by the underlying com- Game on empirical social networks was presented in munication topology. A “speaker” and a “listener”, are Ref. [500]. The initial number of opinions was equal chosen at random. The speaker transmits a word from to the number of interacting agents. The study focuses its vocabulary and sends it to the listener. If the lis- on the impact that communities in the underlying so- tener’s vocabulary contains this word, the interaction is cial graphs have on the outcome of the convergence to- termed “successful”, and both nodes collapse their vo- wards consensus. The main conclusion is that networks cabularies to this one word. Otherwise, the communi- with strong community structure hinder the system from cation is “unsuccessful” and the listener adds this word reaching a global agreement. The evolution of the Nam- to its dictionary. The time unit of the game is the time ing Game in these networks maintains clusters of coex- needed for randomly select N agents as speakers, where isting opinions practically indefinitely. The authors also N is the size of the network. In the context of the spread investigate the agent-based network strategies to facil- of opinion, each node’s vocabulary represents the opin- itate convergence to global consensus. Figure 50 plots ion that the node supports. Therefore, in this context, the number of opinions surviving to the end of simula- the number of opinions is often limited to two, leading tion along the time axis. to a binary version of Naming Game. In such a setting, the agent holding opinion A, and receiving from B his friend opinion B, gets into a mixed state of consid- A ering both opinions. Then next friend interaction will allow this node to select the received opinion as his unique one. This state of hesitation introduces specific resistance to changing the opinion that a person holds immediately upon hearing another one from a friend. The basic properties of the Naming Game were established in Ref. [519]. The authors analyze the role of system size in scaling, the convergence into stable or Figure 50: (A). The number of different opinions Nd versus time for semi-stable states as a function of both the vocabular- different high-school friendship networks and the Watts-Strogatz network. (B). The relative frequency of final configurations with Nd ies of agents and the total number of words generated different opinions for the same high school friendship networks as in by them, the disorder-order transition from the network (A) based on 10,000 independent runs. point of view, and the conditions under which the con- Source: Figure from [500]. 52 The Naming Game model assumes one interaction at is the minimal model with opinion competition. Tab.1 a time instance, and therefore only two different nodes lists all binary model interactions. The dynamics of the are active in the relationship, one as a speaker and an- system with these interaction is represented by the fol- other as a listener. There are some extensions of the lowing mean-field equations: Naming Game to the model groups. For example, there is no restriction on the number of active nodes, or any- dnA = −n n + n2 + n n + 3 pn , one can be both speaker and listener simultaneously. dt A B AB AB A 2 AB (4.1) dnB −n n n n − pn , This variant is suitable for modeling competition be- dt = A B + AB B B tween groups within a peer-to-peer network [521]. As where p is the fraction of the committed nodes in state expected, more intensive interactions enables the global {A} in the population, the fractions of the uncommitted consensus to arise earlier than in the regular Naming nodes in states {A}, {B} in the population are denoted Game. So does also a large number of initial opin- as nA, nB respectively, and the fraction of the nodes in ions within the competing groups. The original Naming mixed state {A, B} is nAB = 1 − p − nA − nB. Game assumes an initial condition in which each agent creates its own word for an observed phenomenon, a Before interaction After interaction natural assumption for the initial applications. How- A A → A A-A ever, in case of opinion spread, limited size vocabulary A or even just binary one could be sufficient for the mod- A → B A - AB A eling spread of opinions with the benefit of simplifying A → AB A-A analysis. Such simplifications are useful for analysis of B Naming Game transition to the stable state. B → A B - AB B B → B B-B 4.1.2. Committed minorities B B → AB B-B The essential change in the convergence time arises A when the committed agents are allowed. Without com- AB → A A-A A mitted agents, the rules of Naming Game endow the lis- AB → B AB - AB tener with a single interaction commitment to its opin- A AB → AB A-A ion. Indeed, receiving the same opinion from two in- B teractions, the agent would change its current opinion AB → A AB - AB B to the new one. The idea of a committed agent is that AB → B B-B his resistance to change opinion extends to the infinite B AB → AB B-B number of interactions with friends sending this agent opinion that it does not hold. In short, committed agents Table 1: Listing of all possible interactions in the binary agreement hold their initial opinion forever. Yet, they are eagerly model. The left column shows the opinions of the speaker (first) and sending it out when selected as speakers. With a cer- listener (second) before the interaction, while the opinion voiced by tain percentage of this type of agent, we could expect a the speaker during the interaction is shown above the arrow. The column on right shows the states of the speaker-listener pair after the consensus change with high probability and high speed. interaction. Numerous studies have shown that this percentage is Source: Table from [497]. significantly below half of the population. That makes it possible for a minority of committed members to en- The fixed-point and stability analyses of the above force a consensus on the majority of society. In Ref. mean-field equations show that for any value of p, the [497], the authors demonstrate that a small fraction, p, consensus state in the committed opinion (nA = 1 − of randomly distributed committed agents can rapidly p, nB = 0) is a stable fixed point of the mean-field dy- reverse the prevailing majority opinion in a population. namics. However, with p below the critical fraction pc, Specifically, when the committed fraction grows beyond two additional fixed points appear; one of these is an un- a critical value pc of about 10% for fully connected stable fixed point (saddle point), whereas the second is graphs, there is a dramatic decrease in the time needed stable and represents an active, steady-state where nA, for the entire population to adopt the committed opin- nB and nAB are all non-zero (except in the trivial case ion. To simplify the analysis, the binary (also called where p = 0). two-word) agreement model us used. In it, only two Figure 51(A) illustrates the impact of the finite net- opinions are defined, A and B. Therefore the binary work size on the steady-state fraction nB of the nodes in model uses only three states: {A}, {B}, and {A, B}. It the state {B} with the critical fraction pc of committed 53 nodes in the state {A} as compared to the mean-field ap- mean-field equations proximation. Figure 51(B) displays a plot of the movement of the stable and saddle points as a function of dnA 2 = −nAnB + nAB + nAnAB + 1.5pAnAB − pBnA, fraction p of committed nodes. f t dn n = −n n + n2 + n n + 1.5p n − p n . f t A B AB B AB B AB A B A B (4.3)

For stylized social networks that the phase diagram of this system in parameter space (pA, pB) consists of two regions (see Fig 52), one where two stable steady-states coexist, and the remaining where only a single stable steady-state exists. These two regions are separated by two fold-bifurcation (spinodal) lines, which meet tan- Figure 51: (A) The steady state fraction nB of nodes in state {B} as a gentially and terminate at a cusp (critical point). function of fraction p of committed nodes in state {A} for complete graphs of different sizes, conditioned on survival of the system. Simulation results are averaged over 100 realizations of the binary agreement dynamics. (B) Movement of the stable fixed point and the saddle point in a phase space as a function of fraction p of committed nodes showing that in this case pc = 0.0957. The inset shows the fraction of nodes in state {B} at the stable (red) and unstable (blue) fixed points as p is varied. Source: Figure from [497].

The time Tc needed to reach consensus can also be computed using a quasi-stationary approximation, which indicates that the survival probability decays exponentially. The precise dependence of consensus times on p can also be obtained for p < pc by considering the rate of the exponential growth of Tc with N. In other words, assuming Tc ∼ exp(α(p)N) yields

γ Figure 52: The phase diagram obtained by integrating the mean-field Tc(p < pc) ∼ exp((pc − p) N). (4.2) equations. The two lines indicate saddle-node bifurcation lines, which form the boundary between two regions with markedly different behavior in phase space. For any values of parameters The simulation results for the case when the underly- within the beak, denoted as region I, the system has two stable fixed ing network topology is chosen from an ensemble of points separated by a saddle point. Outside of the beak, in region II, the system has a single stable fixed point. The saddle-node ER random graphs are also presented in Ref. [497] bifurcation lines meet tangentially and terminate at a cusp bifurcation for the given size N and the given average degree hki. point. The qualitative features of the evolution of the system, Source: Figure from [501]. in this case, are the same as in the case of the complete graph. Having the above result, the natural ques- The simplified binary Naming Game model, in which tion arises about more general cases of opinion evolu- the initial state contains nodes with only two opinions, tion in which more than one group is committed to dis- was studied in Ref. [522]. The authors investigate con- tinct, competing opinions. The simplest case involve sensus formation and establish the asymptotic consen- two groups and two opinions, A and B and they consti- sus times and provide asymptotic solutions for the bi- tute fractions pA and pB of the total population, respec- nary Naming Game. A six-dimensional ODE analyti- tively was studied in Ref. [501]. The authors study the cally captures the dynamics of the binary Naming Game mean-field version of the model. In the asymptotic limit model with committed agents [523]. The authors show of network size, and neglecting fluctuations and corre- that the tipping points for social consensus decrease lations, the system can be described by the following when the sparsity of the network increases. The impact 54 of committed agents on the number of opinions persisting in the network and on the scaling of consensus time is investigated in Ref. [524]. Further investigations of the value of tipping points show that they not only depend on the percentage of minority agents and density of the network connectivity, but also on the distribution of speaker activities over time [525]. The main conclusion is that a group with a higher density of short waiting times between its members speaking activities will enforce a consensus on its opinion over a group with the same overall speaking activity but with a lower initial intensity of speaking. The recent empirical study of the tipping points with the minority of committed agents has been presented in A B Ref. [526]. The authors show that theoretically predicted dynamics of critical fractions do emerge within an empirical system of social coordination based on an artificially created system of evolving social conven- tions. The authors first synthesize the diverse theoretical and observational accounts of tipping point dynamics to derive theoretical predictions for the size of an effective critical fraction of committed agents. A simple model of strategic choice is used in which agents decide which Figure 53: (A). Theoretical modeling of the proportion of outcomes opinion to adopt by choosing the option that yields the in which the alternative behavior is adopted by 100% of the greatest expected individual reward given their history population. In this system, there are N=1000 agents, the number of interactions T=1000 interactions, and M=12 memory to store past of social interactions. The model predicts a sharp transi- interactions used in agent decisions. (B). The value of the predicted tion in the collective dynamics of opinions as the size of critical fraction is shown as a function of individuals’ average the committed minority reaches a critical fraction of the memory length, M. The dashed lines indicate the range enclosed by population (see Fig. 53). Two parameters determine the the experimental trials, showing the largest unsuccessful minority (21%) and the smallest successful minority (25%). Although the theoretical predictions for the size of the critical frac- expected size of the critical fraction increases with M, this tion: the length M of individual memory of the past in- relationship is concave, allowing the predicted tipping point to teractions, and the population size, N. Inspecting these remain well below 50% even for M > 100. Inset shows the effect of parameters shows that the predicted size of the tipping increasing population size on the precision of the critical fraction of committed minority C prediction with M = 12, T = 1000. For point changes significantly, with the individuals’ aver- N < 1000, small variations in the predicted tipping point emerge due age memory length M (see Fig. 53). to stochastic variations in individual behavior. Shaded region The authors recruited 194 subjects via the World indicates sizes of C with which the trials succeed frequently, but without certainty. Above this region, for larger C sizes, the Wide Web, and clustered them into online communi- probability of success reaches 1; for C sizes below this region, the ties for the experiments. Figure 54 shows a summary of likelihood of success goes to 0. final adoption levels across all trials, along with expec- Source: Figure from [526]. tations from the introduced empirically parameterized theoretical model, with 95% confidence intervals. This figure compares these observations to numerical simulations of the theoretical model using population sizes and observation windows comparable to the experimental study (N = 24, T = 100, M = 12). The theoretically predicted critical fraction values from this model fit the experimental findings well. Some of the variability of the critical fraction results shown in Fig. 53 can be explained by one crucial parameter, the strength of the agent’s commitment. This parameter was not controlled in the experiments in Ref. [526]. It was introduced in Ref. [527], where the au- 55 To investigate the sensitivity of Naming Game dynamics for nodes in the multi-opinion state, the following two generalizations were introduced [528]. The first allowing the speaker’s an asymmetric choice of an opinion to send, and the second allowing the listener to keep its mixed state even if the opinion it received is in its state and then studying impact of each of this two changes on the system dynamics. Hence, this version of the Naming Game model gains two continuous parameters. Both parameters vary listener-speaker interactions at the individual level. The first parameter, called propensity, extends the choices of speakers with mixed opinions by defining their probability p , 0.5 of Figure 54: Final success levels from all trials (gray points indicate choosing opinion to send to the listener. The second pa- trials with C < 25%; black points indicate trials with C ≥ 25%) with rameter, stickiness, allows the listener holding a mixed 95% confidence intervals N = 24, T = 45, M = 12 (gray area opinion and receiving matching opinion from the sender indicates 95% confidence for trials with C ≥ 25%). Also shown is to keep its current mixed opinion with certain prede- the theoretically predicted critical fraction (solid line shows results averaged over 1,000 replications). The dotted line indicates fined probability s. The authors use the “listener-only C = 25%. The theoretical model of critical fractions provides a good changing opinion at interaction” version of the Nam- approximation of the empirical findings. For short time periods ing Game introduced in Ref. [529]. The so-generalized (T < 100), the critical fraction prediction is not exact (ranging from 20% < C < 30% of the population); however over longer times Naming Game preserves the existence of critical thresh- (T > 1000) the transition dynamics become more precise (solid line). olds defined by the fractions of the population belong- Source: Figure from [526]. ing to committed minorities. Above such threshold, a committed minority causes a fast (in time proportional to the logarithm of the network size) convergence to thors analyze the impact of the commitment strength consensus, even when there are other parameters, such defined as the number of subsequent interactions with as propensity or stickiness or both, influencing the sys- speakers with opposite opinions needed for the commit- tem dynamics. However, the two introduced parameters ted agent to change the opinion. This strength is infinity cause bifurcations of the system’s fixed points that may for the traditional committed agents. The authors allow lead to changes in the system’s consensus. this strength to be set to any value and change by in- Figure 55 summarizes the most interesting findings. teractions with other agents introducing a waning com- With stickiness, s > 0.5, the new stable region arises in mitment. The results of this analysis show that a shift which a large number of neutrals that are agents hold- in commitment strength affects the critical fraction of ing mixed opinions can effectively block the committed the population necessary for achieving a minority driven agents from reaching consensus. consensus. Increasing the strength lowers critical frac- The experiments on social animals presented in Ref. tion for waning commitment, which can decline via in- [530] confirm these findings. The authors use a group teractions with agents holding the opposing opinion. of cows to demonstrate that, for a wide range of con- Conversely, for the increasing commitment that can ditions, a strongly opinionated minority can dictate strengthen through interactions with agents holding the group choice, but the presence of uninformed individu- same opinion, the critical fraction increases with the als (nodes holding a mixed opinion in terms of the Nam- commitment strength. If the strength of commitment ing Game model vocabulary) spontaneously inhibits is distributed among committed nodes probabilistically this process, returning control to the non-committed according to a distribution, the higher standard devia- majority. The results presented in both references high- tion of such distribution increases the critical fraction light the role of uninformed individuals in achieving for waning commitment, but it decreases this fraction democratic consensus amid internal group conflict and for increasing commitment. Assuming that the partic- informational constraints. Figure 56 shows the com- ipants in the experiments in Ref. [526] have different parison between the final majority reached in a sys- strength of commitment or change their commitment tem without uninformed individuals and with different strength over time might enable the model to find a good numbers of uninformed individuals. In the presence match to the experiment results in Ref. [526] using the of sufficient uninformed individuals, the minority can variable commitment strength from [527]. no longer achieve the majority even by increasing the 56 A A B

Figure 57: A diagram of the modeling, with vertex shapes representing opinions. At each time step, the system with probability φ is updating according to panel (A) and with probability 1 − φ according to panel (B). In (A), a vertex i is selected at random and one of its edges (in this case the edge (i, j)) is rewired to a new vertex j0 that holds the same opinion as i. In (B), vertex i adopts the opinion of one of its neighbors j. Source: Figure from [532]. B Furthermore, properties of the opinions themselves and preferences of committed agents also affect the value of critical fraction. in Ref. [531], assuming that one opinion has a fixed stickiness, the authors investigate how the critical size of the competing opinion required to tip over the entire population varies as a function of the competing opinion’s stickiness. The authors analyze this scenario for the case of a complete-graph topology through simulations and by a semi-analytical approach, which yields an upper bound for the critical minority fraction of the population. In Ref. [502], the authors find that the system with high Shannon entropy has a higher consensus time and Figure 55: Left surface shows the average ratio of consensus time Tc to the network size N as a function of propensity p and stickiness s. a lower critical fraction of committed agents compared Lighter the color is, the longer is consensus time. Both parameters to low-entropy systems. They also show that the critical influence the surface shape. The right panel shows regions of global number of committed agents decreases with the number stability of the system cA-s plane, where cA denotes the fraction of agents committed to an opinion, and s denotes the stickiness to the of opinions and grows with the community size for each mixed opinion. opinion. The most surprising result shows that when the Source: Figure from [528]. number of committed opinion is of the order of the social system size, the critical size needed for minority driven consensus is constant. This means that in such strength of minority preference. a case the critical fraction tends asymptotically to zero when the social system size tends to infinity. This result suggests that committed minorities can more easily enforce their opinion on highly diverse social systems, showing such systems to be inherently unstable.

4.1.3. Agent interaction models Other less popular models of agent interaction and opinion evolution were introduce. For instance, in Ref. [532], the authors present a simple model of opinion evolution in social systems, which is shown in Fig. 57, by combining two rules: (i) individuals form their Figure 56: (A) Without uninformed individuals, as the minority opinions to maximize agreement with opinions of their increases its preference strength, it increasingly controls group neighbors, and (ii) network connections form between opinion. (B) In the presence of sufficient fraction of uninformed individuals, the minority can no longer achieve the majority by individuals with the same opinion. increasing its preference strength. In this opinion system, we would expect the resilience Source: Figure from [530]. to increase over time in the system because the agents 57 are adapting opinions in agreement with the opinions of ever, these conclusions were drawn in experiments in their neighbors. Yet, this will also create latency. How- which both simulations were making random decisions, ever, the dynamics of connection change will reduce the so the results not always were a fair comparison. There- latency by grouping together the like-minded agents and fore in Ref. [535] the authors propose a coordinated thereby speed up the consensus formation. execution of randomized choices to enable precise com- Threshold model. The threshold model focuses on the parison of different algorithms in general. For compari- spread of novelty, in which nodes are in one of two son of sequential seeding with single activation seeding, states: the old opinion, and the new opinion [498]. Here, the new approach means enforcing that when the newly the most crucial challenge is to find the smallest num- activated nodes at each stage of spreading attempt to ber of nodes, called initiators, which starting with the activate their neighbors they use the same random value new opinion would be able to spread this opinion to the to make decision about activating or not the neighbor. entire network. Any node with the old opinion changes Using this approach, the authors prove that sequential it to the new opinion when fraction t of its neighbors seeding delivers at least as good spread coverage as the holds the new opinion. It is similar to the binary Nam- single stage seeding does. Moreover, under modest as- ing Game model. Yet, the semantic of opinion change is sumptions, sequential seeding performs provably better different and one-directional, because the node switch- than the single stage seeding using the same number of ing to new opinion cannot turn back. In the simplest seeds and node ranking. Another interesting result is case of this model, all nodes have the same adoption that, surprisingly, applying sequential seeding to a sim- threshold. The cascade size triggered by a set of ini- ple degree-based selection leads to higher coverage than tiators in this version of the model is a function of the achieved by the computationally expensive greedy ap- initiator fraction [533]. The authors conclude that for proach that was considered to be the best heuristic. Fig- a high threshold, there exists a critical initiator fraction ure 59 compare performance between sequential seed- which, when crossed, assures the global cascade. The ing and greedy heuristic and the theoretical limit. authors also observe that communities can extend opin- A more challenging model in which nodes have dif- ions spread. Different initiator selection strategies are ferent thresholds sampled from the given distribution also studied. of threshold value with known average and variance is The commonly used activation in these models is ac- studied in Ref. [536]. The authors investigate the be- tivating all seeds at once, which achieves the highest havior of cascade sizes in the presence of different num- speed of the spread, but may not gain the highest tran- bers of initiators and various threshold distributions for sition of the network nodes to the new idea. Therefore, both synthetic and real-world networks. The authors ob- several novel approaches for seed initiation rely on a se- serve that the tipping point behavior of the cascade sizes quence of initiation stages [534, 535]. Sequential strate- in terms of the threshold distribution deviation changes gies at later stages avoid seeding highly ranked nodes into a smooth crossover when there is a sufficiently large that have been already activated by diffusion active be- initiator set. The authors demonstrate that for a specific tween seeding stages. The gain arises when a saved seed value of the threshold distribution variance, the opin- is allocated to a node difficult to reach via diffusion. The ions spread optimally. In the case of synthetic graphs, experimental results [534] indicate that, regardless of the spread asymptotically becomes independent of the the seed ranking method used, sequential seeding strate- system size, and that global cascades can arise just by gies deliver better coverage than single stage seeding in the addition of a single node to the initiator set. about 90% of cases. Longer seeding sequences tend to Voter model. The voter model is likely the simplest activate more nodes but they also extend the duration of model for the spread of opinions with dynamics that diffusion. Figure 58 shows that various variants of se- may lead to consensus [537]. Each node in the under- quential seeding resolve the trade-off between the cov- lying network selects a single opinion from the two op- erage and speed of diffusion differently. posing opinions allowed in the model. Starting from a The results show that sequential seeding is nearly random initial distribution of opinions, the model dy- always better than its single stage equivalent with the namically evolves in steps. Each step selects a random same parameters. The global results for all networks, node without opinion and assigns it an opinion of one strategies and parameters show better results than se- of its nearest neighbors also chosen at random. The dy- quential seeding in 95.3% of simulation cases. The re- namics are simple here; the opinion with the initial ma- sults from simulations demonstrate that the improve- jority is likely to bring all nodes to the consensus on this ment can also exceed 50% with the use of the same opinion. number of seeds as in the single stage seeding. How- An exciting extension of this model enables the node 58 Figure 58: Balancing the speed and the coverage for single stage and sequential seeding. (A) k per stage sequential seeding strategy (SN kPS) with k = 1 compared with the single stage approach (SN). (B) Sequential strategy based on the reference time (SQ TSN) compared with the single stage approach (SN). (C) Sequential strategy with revival mode (SQ TSN R) compared with the single stage approach (SN). Source: Figure from [534].

drastic consequences for the global behavior of opinion formation models in the case of dynamically evolving networks. The mean-field analysis accounts for differences due to the asymmetric coupling between interacting agents and the asymmetry of their degrees. The ne- cessity for agents to transit via an intermediate state before changing their opinion strongly enhances the trend towards consensus. Allowing the interacting agents to bear more than one opinion at the same time drastically changes the model’s behavior and leads to fast consen- Figure 59: Performance comparison on undirected networks. (A) sus. The averaged performance of the sequential SQ and single stage SN seeding using greedy heuristic for node selection. The results are Axelrod model. An interesting variant of the Axelrod shown as a fraction of the maximum coverage CMax and as a model [538] was introduced and studies in [539]. The function of the network size N, probability of propagation PP across each edge, and the fraction of nodes selected as seeds (seed selection authors study a variant of the Axelrod model on a net- percentage) SP. (A1) Performance of the sequential SQ and single work with a homophily driven rewiring rule imposed. In stage SN seeding using the degree based ranking. The results are this model, network nodes represent individuals. A set compared with maximum coverage and the upper bound as a of F independent attributes defines every node’s state. function of the individual configurations, each defined by N, PP, SP. To achieve the coordinated execution, the random binary choices for Each attribute can take one of q distinct traits repre- each edge to propagate or not information across this edge are made sented by integers in the range [0, q − 1]. Initially, each at simulation initialization and applied to all compared executions. attribute of each node randomly chooses its value from (B) Coverage of the sequential method SQ as percentage of CMax q integers. placed between the single stage SN and Max using greedy nodes selection; (B1), (B2) Sequential seeding performance SQ plotted The network is interconnected randomly as an Erdos- between single stage seeding SN and Max using random and Renyi (ER) random graph with the given average degree degree-based node selections, respectively; (B3) Performance of sequential SQ and single stage seeding SN represented by percentage hki. Each time step randomly selects a node i and then of activated nodes within the network (coverage) for random seed its neighbor j. Then, the similarity between nodes i and selection, degree-based ranking and greedy seed selection in j is computed by counting the number of attributes for comparison with the maximum coverage as a function of the i j individual configurations. which and possess the same trait. If the similarity is Source: Figure from [535]. equal to or above the given threshold, the influence step is executed. In it, node j adopts the trait of node i for the randomly chosen attribute from those for which nodes i not only to update its opinion but also to break and es- and j currently do not agree. Otherwise, the rewiring tablish connections with other nodes [496]. The au- step dissolves the link between i and j and replaces it thors also introduce further model variations, such as by a link from i to randomly selected node k from those “direct voter model” or “reverse voter model”. They not already connected to i. The authors show that with also use the memory-based Naming Game style opin- these dynamics and in the presence of committed agents ion change in which no agent can move directly from whose fraction exceeds the critical value, the consensus one opinion to another, but needs to transit via an inter- time becomes logarithmic in the network size N. More- mediate state. The so-introduced mechanisms are stud- over, the author also demonstrates that slight changes ied using a mean-field analysis. The authors show that in the interaction rules can produce strikingly different slight modifications of the interaction rules can have results in the scaling behavior of consensus time, Tc. 59 However, all the interaction rules tested in the study resource by pheromone marked trail, and the maximum qualitatively preserve the benefits gained from the pres- rate at which an ant can leave the trail. ence of committed agents. This leads to cubic equation for x and to equilibrium solutions at which dx/dt = 0. Setting s = 10, and us- 4.2. Survive resilience in social animal systems ing two value of alpha 0.021 and 0.0045 results in plots Analogous to the social systems of humans are the shown in Fig. 60. social systems composed of social animals (ants, hon- eybees, etc.). Those are also within the scope of this A B chapter. Most often, the studies in this area include the foraging behavior [540, 541], nest building [542], and copying with stresses [543, 544]. The resilience of social insects depends on three crucial elements of such system infrastructure: transportation networks, supply chains, and communication networks [545]. To assure resilience, the system may use three different pathways: resistance, redirection, and reconstruction. Figure 60: The predicted total number of ants, x, walking to a single The human-infrastructure networks currently un- food source as a function of colony size, n, when the feeder is found (A) frequently (α = 0.021) and (B) infrequently (α = 0.0045) by dergo rapid decentralization and increased interconnec- independently searching ants. The other parameters, β = 0.00015 and tivity. Hence, interesting analogies arise between them s = 10, are fixed. Blue line shows the theoretically unstable points. and the social insect infrastructures. Those analogies Red line shows the theoretically stable points. Black dots represent can inform management of human infrastructure re- the results from the simulation. Source: Figure is modified from [540]. silience from social insect infrastructure resilience design. Inversely, researchers of social insects may benefit from adopting the sophisticated analytical and sim- The authors observe hysteresis arising for low inde- ulation tools developed for the study of human infras- pendent searching ability α and intermediate values of tructure resilience. The essential difference between the colony sized n. This directly demonstrates that forag- two design strategies is that unlike in human social sys- ing in Pharaoh’s ants is influenced by colony size in a tems, the change in animal social systems [540, 544] nonlinear and discontinuous way. This transition in for- can come only via evolving the particular species social aging is strongly analogous to the first-order transition dynamics. in physical and social systems. The theoretical modeling of the honeybee’s foraging 4.2.1. Social foraging and nest building dynamics in one-source and multi-source settings is presented in Ref. [541]. The eventual number of foragers The study of the foraging behavior in ant colonies depends on the bee concentration and the scouting rate. that focuses on transition between disordered and or- By fixing the scouting price, the author can analyze the dered foraging in the Pharaoh’s ant is presented in Ref. dynamics of the proportion of potential foragers forag- [540]. The authors found that small colonies forage in a ing in cases of the one-source and two-source settings. disorganized manner, while large colonies transition to The results show that both phase transitions and bista- organize pheromone-based foraging. bility arise in the proposed model. The simulation re- The theoretical model derived from those empirical sults for the one-source case are shown in Fig. 61. findings is defined as follows: The collective process of ant nest building is studied in Ref. [542]. To analyze the shape diversity of dx dt = (ants beginning to forage at feeder) the nests, the authors conduct two-dimensional nest- −(ants losing pheromone trail) (4.4) digging experiments under homogeneous laboratory sx = (α + βx)(n − x) − s+x , conditions in which the shape diversity emerges only from digging dynamics. A stochastic model highlights where variable x is the total increase of the ants walk- the central role of density effects in shape transition. ing to a single food source. Variable n stands for the Figure 62 shows the experiment results. The digging colony size. Constants α, β, and s denote respectively dynamic follows the equation: the probability per minute per individual ant that she finds food through independent searching, the probabil- A tα A = M , (4.5) ity per minute per individual that she is led to the food βα + tα 60 from calm to violent states in response to thermal stress. The evaluation of modeling is done by subjecting experimental societies of the spider Anelosimus studiosus to heat stress. The authors demonstrated that both colony size and members’ personality compositions influence the timing of and recoverability from sudden transitions in the social state. Figure 63 illustrates the developed three-state model for social tipping points in social spiders.

∗ Figure 61: Attractors and dynamics of X /C (a rate of the stable B number of forages X∗ to the actual number of foragers C) for one-source bee foraging systems. f = 1, α = 0.01. Blue line shows the theoretically unstable points. Red line shows the theoretically stable points. Black dots show the results of the simulation. Source: Figure is modiﬁed from [541].

C where A (in cm2) is the excavated area (i.e., the nest area), t (in hours) is the time elapsed since the start of nest digging, AM is the maximal area of the nest that is ultimately dug out, α stands for the cooperation level between ants, and β is the time value when A = 0.5AM.

Figure 63: (A). The authors developed a three-state model describing the interaction of docile and aggressive spiders, which allowed for transitions of agitation state among aggressive spiders. (B). For a six-spider colony (N = 6), the model predicted one, two or three Figure 62: Example of experimental results. The gray dashed vertical equilibria as a function of temperature. (C). An example when line corresponds to the occurrence of the morphological transition, hysteresis appearance depends upon colony composition. (D). An which separates the first and second growth stages. Evolution of the example when colony size determines the shape of hysteresis. nest area and digging rate plotted with the fitting parameters: Source: Figure from [543]. 2 2 AM = 218.10cm ; α = 1.95; β = 16.3h; r = 0.99. The gray line represents the area, the black dashed line corresponds to the fitting curve, and the black curve represents the digging rate. The digging Experiments and field studies were combined in Ref. dA rate is calculated as the derivative dt of the fitted area. [544] to investigate the social and ecological factors af- Source: Figure from [542]. fecting cold tolerances in range-shifting populations of the female-polymorphic damselfly Ischnura elegans in northeast Scotland. The range-shifting, i.e., the pole- 4.2.2. Social animals response to stress ward shifts in geographical ranges, happen when the The factors contributing to the appearance of abrupt climate warms, and the geographical position of some tipping points in response to stress in animal societies species thermal range limit changes with it, and they are studied in Ref. [543]. The authors first constructed expand their range to fill these new thermal boundaries. an analytical model of how the personality compositions The authors also consider the consequences of chang- of society members could alter their propensity to shift ing cold tolerance for evolutionary change. Both envi- 61 ronmental and social effects on cold tolerance and female color morph frequencies were recorded. Density manipulations in the laboratory provided experimental evidence that social interactions directly influence cold tolerance. The authors suggest that there is a broader need to consider the role of evolving social dynamics to shape both the thermal physiology of individuals and the thermal niches of their species reciprocally. Fig- ure 64 shows the results of the study.

Figure 65: Potential responses of a system to a disturbance. System performance is an experiment-specific measure of system functionality (harvesting rate, brood production rate, traffic flow). Time t0 is the start of the experiment, pre-disturbance, td indicates the start of the disturbance and tr indicates the beginning of the recovery phase for a system while tc indicates the point at which recovery is complete since system performance has returned to pre-disturbance levels. (A) Shows a system that has invested in resistance and as a result does not experience a decrease in functionality after the disturbance. (B) Shows a system that is using redirection. Although there is an initial decrease in performance, it is rapidly mitigated by rerouting flows using existing infrastructure. (C) Shows a system that uses primarily reconstruction-based resilience strategies. Since reconstruction requires the construction of new infrastructure, it takes longer to recover pre-disturbance performance. (D) Shows a non-resilient system, which does not recover pre-disturbance performance. Source: Figure from [545]. Figure 64: Environmental (A-C) and social (D-F) determinants of cold tolerance in adult and larval Ischnura elegans captured near their elevation range limit in Scotland. The x-axis “Habitat authors start with listing concepts directly related to so- suitability” is computed from a maximum entropy model for species distribution [546]. With higher habitat suitability, the model predicts cial tipping points, including behavioral states and en- the area to be more suitable for living. The “Density treatment” in vironmental parameters, attractors and basins of attrac- (F) means different scenarios of social pressure – whether being tion, as well as perturbations. Among the social proper- placed in the environment alone or with another member. ties most relevant to tipping points, they list relatedness Source: Figure from [544]. and group size, keystone individuals, behavioral diversity, social organization, and prior experience. Finally, The resilience of three critical social insect infrastruc- among the types of social tipping points, they list social ture systems: transportation networks, supply chains, scale, metabolic tipping points, and social or cognitive and communication networks is reviewed in Ref. [545]. tipping points. The authors describe how systems differentiate investment in three pathways to resilience: resistance, redirection, or reconstruction (cf. Fig. 65). The authors 4.3. Resilience of the planetary social-ecological sys- also observe that human-infrastructure networks un- tem dergo rapid decentralization and increased interconnec- Socio-ecological systems, also known as Human- tivity, thus becoming more like social insect infrastruc- Environmental Systems (HES’s), Socio-Ecological Sys- tures already are. Therefore, human infrastructure man- tems (SES’s), or “coupled human-and-natural systems” agement might learn from social insect researchers. In have been widely studied recently. Such systems are turn, the latter can make use of the mature analytical ubiquitous in agriculture, water use, terrestrial and and simulation tools developed for the study of human aquatic systems, the global climate system, and else- infrastructure resilience. where [58]. A social-ecological analysis of resilience A review of the tipping points arising in the dynam- enables the study of people-environment interactions ics of animal societies is provided in Ref. [547]. The across varying dimensions, times, and scales [548]. 62 Serious attempts to integrate the social dimension into such studies are currently most often associated with works focusing on resilience. The large numbers of sciences involved in such explorative attempts led to the discoveries of new links between social and ecological systems. Recent advances include the understanding of social processes, like social learning and social memory, mental models and knowledge-system integration, visioning and scenario building, leadership, agents groups, social networks, institutional and organizational inertia and change, adaptive capacity, transformability and systems of adaptive governance that allow for management of essential ecosystem services [62]. Integrating the natural science and the social science Figure 66: The core subsystems in a framework for analyzing aspects within one framework has been a crucial chal- social-ecological systems. Source: Figure from [556]. lenge for such studies. Whereas a “system” is practically a universal concept in the natural sciences, they vary across social sciences and their branches, placing while undergoing change so as to retain necessary func- them outside the realm of natural sciences. tion, structure, identity, and feedback) has four compo- One possible approach to this challenge is presented nents: latitude, resistance, precariousness, and panar- in Ref. [549]. The authors suggest using an institu- chy, the most readily portrayed using the metaphor of a tional lens to integrate social and natural aspects of the stability landscape [557]. joint resilience studies because institutions may become The origin of the resilience perspective and an methodological linchpins for integrating the social and overview of its development to date are presented in the natural sciences for the sake of sustainability. More- Ref. [62]. With roots in one branch of ecology and the over, the progress in social sciences and core under- discovery of multiple basins of attraction in ecosystems standing of social system evolution enables researchers in the 1960-1970s, it inspired social and environmental to evaluate more formally than in the past, the influence scientists to challenge the dominant stable equilibrium of different institutions on the social-ecological system view. resilience. For example, global and international insti- The institutional configurations that affect the inter- tutions can promote and support building collaborations actions among resources, resource users, public infras- among countries to enhance resilience by negotiating tructure providers, and public infrastructure are dis- global agreements and treaties [550, 551]. The national cussed in Ref. [558]. We summarize the results and local governments can support the organizational in Fig. 67 that illustrates the framework, Tab.2 that resilience of industries to strengthen the social response presents the entities involved, and Tab.3 that depicts the to social unrest [552]. The resource-dependent indus- actual links involved. The authors propose a framework tries can support the resilience of social systems by sus- that helps to identify SES’s potential vulnerabilities to taining their workforce in times of social stress[553]. disturbances. The authors posit that the link between re- Labor market resilience may increase the resilience of source users and public infrastructure providers is key to social systems against recession. Economic resilience affecting the robustness of SES’s even though this link may increase the resilience of social systems against has frequently been ignored in the past. disasters [554, 555]. The intellectual roots and core principles of social Sustainability is a topic often studied jointly with re- ecology are traced in Ref. [548]. The authors demon- silience in the context of SES. in Ref. [556], the au- strate how these principles enable a broader conceptu- thors have shown a general framework to identify ten alization of resilience that may be found in much of the subsystem variables that affect the likelihood of self- following literature. organization in efforts to achieve a sustainable SES (see Fig. 66). Three related attributes of social-ecological systems 4.3.1. Early warning signal in social-ecological system (SES’) determine their future trajectories: resilience, A coupled HES that is close to a tipping point is gen- adaptability, and transformability. Resilience (the ca- erally far less resilient to perturbations than the same pacity of a system to absorb disturbance and reorganize system far from such points [58]. The authors of [58] 63 Entities Examples Potential Problems A. Resource Water source Fishery Uncertainty Complexity / Uncertainty

B. Resource Users Farmers using irrigation Stealing water, getting a free ride on maintenance Fishers harvesting from inshore ﬁshery Overharvesting

C. Public infrastructure providers Executive and council of local users? association Internal conﬂict or indecision about which policies to adopt Government bureau Information loss

D. Public Infrastructure Engineering works Wear out over time

Institutional rules Memory loss over time, deliberate cheating

External Environment Weather, economy, political system Sudden changes as well as slow changes that are not noticed

Table 2: From [558]. Entities involved in social-ecological systems

where 4U is the utility gain of changing opinion, x is the function of time representing the proportion of the population adopting the opinion, k is the social learning rate; 3. The coupling is represented by equations

 h(1 − x)F  F˙ = RF(1 − F) − ,   F + s (4.8)  1  x˙ = kx(1 − x)[d(2x − 1) + − ω], F + c where c is the rarity valuation parameter, d is the Figure 67: A conceptual model of a social-ecological system. social norm strength, and ω is the conservation Entities are defined in Table.2 while links are detailed in Table.3. cost. Source: Figure from [558]. The early warning signal approaches show the potential for warning of social-ecological regime shifts [559]. show also that the coupled HES can exhibit a richer va- They could be valuable in natural resource management riety of dynamical regimes than the corresponding un- to guide management responses to variable and chang- coupled system. Thus, early warning signals can be ing resource levels or changes in resource users. How- ambiguous because they may herald either collapse or ever, investigation of specific cases of social-ecological conservation. Moreover, the authors also observe that regime shifts is required to make sure that, first, the re- human feedback can partially mute the early warning quired data are available for the studied cases and, sec- signal of a regime shift or cause the system to evolve ond, the resulting early warning signals are robust. The toward and perpetually remain close to a tipping point. third and fundamental criterion by which to evaluate an The coupling of the environment dynamics models early warning signal in a specific case study is whether and the social dynamics model presented in Ref. [58] the signal can give sufficiently early warning for the can be summarized as follows: transition to be avoided. Fig. 68 shows the results of such analysis. 1. The forest ecosystem is modeled as A theoretical framework is built in Ref. [560] to dF hF demonstrate that rising variance-measure, for example, = RF(1 − F) − , (4.6) dt F + s by the maximum element of the covariance matrix of the network, is a valid leading indicator of a system ap- where R is the net growth rate, s is the supply and proaching instability. The authors show that this indi- demand parameter, h is the harvesting efficiency cator’s reliability and robustness depend more on the while F is a function of time representing the size pattern of the interactions within the network than the of the forest; network structure or noise intensity. Mutualistic, scale- 2. The social dynamics model is defined as free, and small-world networks are less stable than their dx antagonistic or random counterparts are, but this leading = kx(1 − x)4U − (−kx(1 − x)4U), (4.7) dt indicator more reliably predicts their instability. 64 Link Examples Potential Problems (1) Between resource and resource users Availability of water at time of need/availability of Too much or too little water/too many uneconomic fish fish? too many valued fish

(2) Between users and public infrastructure Voting for providers Contributing resources Rec- Indeterminacy/lack of participation Free riding providers ommending policies Monitoring performance of Rent seeking Lack of information/free riding providers

(3) Between public infrastructure providers and Building initial structure Regular maintenance Overcapitalization or undercapitalization public infrastructure Monitoring and enforcing rules Shirking disrupting temporal and spatial patterns of resource use Cost/corruption

(4) Between public infrastructure and resource Impact of infrastructure on the resource level Ineﬀective

(5) Between public infrastructure and resource dy- Impact of infrastructure on the feedback structure of Ineﬀective, unintended consequences namics the resource? harvest dynamics

(6) Between resource users and public infrastruc- Coproduction of infrastructure itself, maintenance No incentives/free riding ture of works, monitoring and sanctioning

(7) External forces on resource and infrastructure Severe weather, earthquake, landslide, new roads Destroys resource and infrastructure

(8) External forces on social actors Major changes in political system, Conﬂict, uncertainty, migration, greatly migration, commodity prices, and regulation increased demand

Table 3: From [558]. Links involved in social-ecological systems.

4.3.2. Regime shifts in social-ecological systems cal results and analytical estimations verify this conclu- The regime shifts in social-ecological systems are sion. Interestingly, the hysteresis effect is more reliable discussed in Ref. [58]. The authors observe that the in a scale-free network than in a random network, even theoretical models of social-ecological systems are re- when both networks have the same average degree. ceiving growing attention. Hysteresis transitions have A social-ecological regime shift in a model of har- been observed in coupled social-ecological systems on vesters of a common-pool resource is studied in Ref. networks. Complex community structure in social- [559]. The authors find that such a change may avoid ecological systems can create multiple small regime over-exploitation of the resource by social ostracism shifts instead of a single large one. Human feedback in of non-complying harvesters. The authors use the ap- a social-ecological system can be fundamental to shap- proach of generalized modeling to study the robustness ing regime shifts in environmental systems. Because of of the regime shift to uncertainty over the specific forms nonlinear feedback, resource collapse can be caused by of model components. The authors introduced their surprising and counter-intuitive changes. Thus, failing generalized modeling using a simple example of a sin- to account for human feedback can underestimate the gle population X that can increase due to a gain pro- potential for regime shifts in ecological systems. cess G(X) and decrease due to a loss process L(X). A generalized model, in differential equation form, for the Four modeling approaches are explored in Ref. [561] population X is and applied to the study of regime shifts in coupled dX socio-environmental systems. They are statistical meth- = G(X) − L(X). (4.9) ods, models of system dynamics, models of equilibrium dt points, and agent-based modeling. A set of criteria has The corresponding Jacobian Matrix J of Eq. 4.9 con- been established to (1) capture feedback between social sists of the single element which is also the eigenvalue and environmental systems, (2) represent the sources of λ 0 0 regime shifts, (3) incorporate complexity aspects, and J = λ = G ∗ − L ∗. (4.10) (4) deal with regime shift identification. The additional parameters are introduced to relate the The role of social networks in social-ecological model to the real world scenarios: scale, elasticity and regime shifts is investigated in Ref. [562]. The authors ratio parameters. By analyzing the Jacobian Matrix J also study the corresponding hysteresis effects caused and referring to contextual knowledge, one could iden- by the local ostracism mechanism under different so- tify and evaluate possible bifurcations. cial and ecological parameters. The results show that Figure 69 shows the actual model considered in Ref. lowering of network nodes degrees reduces the hystere- [559]. The corresponding generalized model can be sis effect and also alters the tipping point. The numeri- written as follows: 65 R A B 55 0.8 50 0.6 45 40 0.4 35 0.2 �.------< 0.02

0.01

0 0.00

-1 -0.01

-2 -0.02 0.76 -,....------1 autocorrelation autocorrelation 0.74 0.84 Kendall tau= 0.179 0.72 0.82 0.70 Kendall tau=-0.542 0.68 0.80 0.66 0.78 0.64 0.0085 0.70 standard deviation standard deviation 0.68 Kendall tau= 0. 164 0.0080 0.66 0.0075 Kendall tau= 0.408 0.64 0.0070 0.62 0.0065 0.60 0.0060 0 100 200 300 400 0 100 200 300 400 lime time

C D q -0.09 eigenvalue GM Kendall tau= 0.870

-0.12 0 represent state variable, ovals show intermediate quantities, and q cumulative distribution 0 explosion symbols represent sources and sinks of ﬂows. 0 100 200 300 400 -1.0 -0.5 0.0 0.5 1.0 Source: Figure from [559]. time Kendall tau

Figure 68: Early warning signals. (A-B). Time series in the lead-up to a regime shift of the model (black line, for parameters of the simulation see text) with filtered fit (red line, only over the time range prior to the regime shift to be used in the following analysis); detrended fluctuations; and their autocorrelation and standard deviation. (C). Generalized modeling-based early warning signal preceding the regime shift. (D). Cumulative distributions of the Kendall τ statistics for the different warning indicators (GM = generalized modeling-based signal; ac = autocorrelation; sd = standard deviation) over 1,000 simulations of the regime shift with different noise realizations. All Kendall τ statistics shown were calculated over the time range 300 to 377. Source: Figure from [559]. Figure 70: Surface of fold bifurcations for ranges of generalized parameters matching. Source: Figure from [559]. dR = c − D(R) − Q(E( f ), R), (4.11) dt c where c is the resource inflow or growth rate (which is bifurcation diagrams of simulation models on paired- independent of the current resource level), D(R) is the parameters. Two developments are identified in Ref. natural resource outflow rate or mortality and Q(E, R) is [437] that enable users to build a framework for un- the resource extraction. Here, E( fc) is the total effort ex- derstanding systemic resilience. The first one is es- erted by the harvesters, which decreases with increasing tablishing dynamical indicators of resilience, i.e., early proportion of co-operators. The changes in the fraction warning indicators. The second is increasingly ubiq- of co-operators are represented by the following equa- uitous collection of the needed data on the dynamic tion: time series for humans and livestock enabled by the rapid rise of technologies for automated recordings, especially wearable electronics. The authors also suggest d fc = fc(1 − fc)(−F(E( fc), R) + W + ω( fc)). (4.12) that humans and the livestock that they consume may be dt seen as a complex network with strong impact on ecol- Figure 70 plots the surface of fold bifurcations for ogy. The dynamical indicators for this network may ranges of generalized parameters. Figure 71 shows the be estimated directly from the interactive dynamics of 66 thors point out some of the difficulties that currently A B weaken the feasibility of studying resilience in social systems. The most important among them are the following.

• The ontological diﬀerences (e.g., social scientists tend to study resilience on the individual level C D rather than on the community level considered in ecological studies). Researchers are reluctant to use system as an ontological description of society.

• The boundaries between disciplines have been constructed already, although the use of institutional lenses following the tradition in social sci- E F ence studies could help with this issue.

• Social-ecological systems exhibit thresholds that when exceeded, result in changed system feedback that leads to changes in function and structure, thus the analogy of the ball and the undulating surface G H is problematic in relation to social phenomena. • As self-organization is becoming an overriding or- ganizing principle in complexity theory in which resilience theory is rooted, its mechanism is dubi- ous in the social system setting.

• The similarities between resilience theory and abandoned theories of functionalism, which are now being replaced by theoretically stronger theo- Figure 71: Bifurcation diagrams of simulation models. ries for understanding of society, are raising doubts Source: Figure from [559]. among social scientists about applicability of resilience theory to their domain. its nodes. Such approach may help tip social and ani- In addition to the inherent complexity introduced by mal dynamic models from a reductionism to a systemic the nature of social system problems, scientists could paradigm. not bypass the difficulty of coupling multiple systems Stability of planet ecology is directly impacted by the when studying a single social system resilience phe- growing human population and its food eating habits. nomenon. As pointed in Ref. [563], one of the threats is the in- The interdependence of the regime shifts in ecosys- creasing meat consumption, which requires high energy tems is studied in Ref. [185]. The authors construct input compared to alternatives. Kanerva [563] discusses a weighted directed network of 30 regime shifts from a how the societies can voluntarily lower meat consump- database containing over 300 case studies based on a lit- tion. erature review of over 1000 scientific papers. The links in this network represent causal relationships (having a 4.3.3. Difficulties in the studies of resilience in social cascading effect or simply sharing standard drivers) be- networks tween each link endpoints. The analysis shows that the The integrated research efforts for sustainability cascading effect accounts for ∼ 45% of all regime shift driven by the urgency of addressing the climate change couplings analyzed, implying structural dependence. threat are analyzed in Ref.[549]. They start with a de- The key lesson from this study is that regime shifts can tailed review of the core concepts and principles in the be interconnected, and they should not be studied in iso- resilience theory that could cause disciplinary tensions lation under the assumption that they are independent of between the social and natural sciences. Then, the au- each other. It is valid for resilience study in general and 67 indicates we should always include all necessary con- On the other hand, critical infrastructures usually are text when studying social system resilience to achieve complex and contain a lot of non-linearly coupled com- precise modeling. ponents, which may adapt and learn from the changing environments. Consequently, the CI system, as a typical complex adaptive system, is more than the sum of 5. Failures in critical infrastructure (CI) systems its parts [566]. The complexity of CI lies in its emergent properties, as a result of interaction and synergy in Critical infrastructures (CI), are the backbone of hu- a large-scale space and time. Therefore, CI systems can man society and play an essential role in transporting adapt and absorb uncertain disturbances while retaining materials and services between distant locations. These their basic functionalities. This ability is called “system critical infrastructures span a multiple scales of space resilience” (also referred to as “ecological resilience” as and time to provide various flow services. Examples in- discussed in Chapter2), which is di fferent from the “en- clude power grid, gas and petroleum systems, telecom- gineering resilience,” a system’s ability to bounce back munication, banking and financial systems, transporta- after perturbations or stresses. Given the unexpected tion, water distributions, and emergency management circumstances and interdependence, it becomes increas- systems. The power grid is a critical infrastructure, yet it ingly urgent to understand and improve the resilience of often breaks down due to unexpected large-scale black- critical infrastructures. Resilience brings the ability of outs, which has been attracting high research interests. adaptation, absorption, and recovery from various per- The Internet symbolizes the beginning of the informa- turbations, faults, attacks, and environmental changes. tion era and has become one of the fast-growing criti- The study for system resilience stems from the self- cal infrastructures. There are also many other critical recovery of ecosystems [4] (also see Chapter2) and bi- infrastructures embedded in different industry sectors. ological systems [380] (also see Chapter3). An ecosys- Due to their crucial role for society, partial or complete tem may automatically restore itself from the species dysfunction of the critical infrastructure leads to large- invasion or environmental changes, and a cellular net- scale damages. work can automatically recover by changing the expres- While serving as the lifeline or backbone of the whole sion level of some specific genes. Since C. Holling and society, critical infrastructures may suffer various risks his colleagues raised the concept of resilience in ecosys- of perturbations or malicious attacks. On the one hand, tems [4], the resilience theory has been generalized not under specific scenarios, small faults in the localized re- only to biology [254] but also to climate [219], eco- gion will accumulate and may propagate in a domino nomics [567], and social systems [568] (also see chap- effect way, leading to lethal cascading failures. For ex- ter4). It suggests a new possible direction for system ample, a line trip caused cascading failures in the power reliability management. For critical infrastructures and grid of North America in 2003, with damage around 10 other complex engineering systems, resilience engineer- billion dollars. According to the North American Elec- ing has recently become widely known to its research trical Reliability Council (NERC) [564], outages affect community [569]. nearly 700,000 customers annually. Under the most tor- rential rain on July 21 of 2012, from the last 61 years, 5.1. Structural properties of CIs city-scale traffic breakdown has happened in Beijing, Research of system resilience relies on our under- which caused economic losses of 11.6 billion yuan. Ac- standing of failures, which is related to but differs from cording to statistics, traffic congestion costs Americans the system’s reliability and vulnerability. Traditional re- 1 $166 billion annually . The eruption of the Icelandic liability analysis mostly follows a reductionism concept, volcano in 2010 has led to the cancellation of at least assuming that the system failures are composed of cer- 60% of daily European flights, resulting in the break- tain failure combinations of a set of components [570]. 2 down of airline networks . Besides extreme weather, The failures of different parts have no or weak correla- over the internet, malicious cyber-attacks have been oc- tions, missing the possible strong dependency between curring ever more frequently [565], increasingly threat- components. However, components are not isolated but ening to paralyze the whole server systems. interconnected with one another, forming a complex network. A network is a graph composed of nodes and links that connect the nodes. For the transportation sys- 1 https://www.truckinginfo.com/338932/traffic-congestion-costs- tems, the road intersections represent nodes, and roads americans-166-billion-annually 2The Sydney Morning Herald., http://goo.gl/gvy9Kg, (2010) connecting two intersections are links. For the power (Date of access:04/04/2014). grid, generators or transmitters constitute nodes, and 68 transmission lines are links. For the Internet, routers mension characterizes the spatially embedded networks are nodes, and cables are links. In this way, failures of with 0 < δ < ∞. It plays a central role, not only in the CI system are described and seen as emergent be- characterizing the structure but also in determining the haviors understood as a collective result of interactions dynamical properties on the network and its behavior between different hierarchical levels. A series of works near a critical point. about technical complex networks, including CI, analyze them from the point of view of complexity science 5.1.2. Small-world connectivity of CIs [571, 572, 573, 41, 574]. The critical infrastructure net- Topologically, small-world networks usually have works usually contain numerous components coupled small distances and high clustering coefficient. Exam- by nonlinear dynamical interactions. Their structure ples of CI in such category are road networks [587], properties, discussed below, determine the systems’ re- railway networks [588], the worldwide maritime trans- silience [575, 576], and its aspects such as the spatiality, portation network [589], and power grid [590]. The small-world property, modularity, rich clubs property, phenomenon of small-world was first discovered in a and more. letter-delivering experiment [591, 592], where the distance between two people in the world was found to 5.1.1. Spatial aspects of CIs be 5.2 on average. A network model developed in Although many complex systems, including social or 1998 [590] explains the formation of a small-world net- biological systems, exhibit common topological proper- work through randomly rewiring of p fraction of links ties, the complexities of CI networks differ from other of a regular lattice. For small p, the systems show new complex networks. For example, the degree distribu- properties of short global path length and high local tions are found to be scale-free for some networks, such clustering coefficient, which rarely appears in the orig- as the gene regulatory networks in biology. Many CI inal regular lattice. According to the statistical proper- networks are embedded in spatial space with certain ties, small-world networks could be classified into dif- constraints, resulting in degree distributions usually not ferent types with different scale constraints [593]. La- being scale-free [577]. Good example of such con- tora et al. [594] introduced the concept of network straints are road network [578, 579]. With the renormal- efficiency as a measure of how efficiently communi- ization procedure, some complex networks are found cation networks exchange information and show that to have self-similar structures, measured with fractal small-world networks are globally and locally efficient. dimension [580]. CI networks usually have geomet- Kleinberg has also demonstrated that one can find short ric constrains and are embedded in a two- or three- chains effectively with pure local information in small- dimensional (3D) space [581]. We call these networks world networks [595]. “spatial networks” [582], such as 2D or 3D lattices. Dynamically, CI networks also have small-world The network dimensionality is fundamental since it not properties [596]. For instance, in urban traffic, two dif- only determines how the system is connected in differ- ferent modes of critical percolation behaviors alterna- ent scales but also has important implications for system tively appear in the same network topology under differ- functions and failures. For a d-dimensional regular lat- ent traffic dynamics. During the rush hours, it shows the tice with N nodes, this average shortest path hli scales properties of the critical percolation of the 2D lattice. as hli ∼ N1/d. Actually, spatial networks usually have a During the non-rush hours or days off, it shows simi- few long-range connections to bridge distances between lar percolation characteristics as small-world networks. far away areas to achieve the balance between optimal It is due to the presence or absence of high-way roads transportation efficiency and costs. Examples are the in a metropolitan city, which behaves as the long-range city highways in the road networks or the high-voltage connection in the classical small-world network model transmission lines in the power grids. With the power- [590]. law distribution of link length, spatial networks could Based on the evidence of both topology and dynamics have dimensions higher than those that are embedded of the transportation network, resilience is not purely a in space [583], which may suggest the existence of hy- structural problem. Indeed, consider traffic in the trans- perbolic space [584]. For example, in a mobile phone portation system, which can become gridlocked even if communication network, the probability P(r) to have a a natural disaster leaves the underlying roads, bridges, friend at distance r decays as P(r) ∼ r−δ, where δ = 2 and railways relatively unscathed. There is mounting [583, 585], and in the global airline network, the proba- recognition that to truly address the resilience of net- bility that and airport has a link to an airport at distance worked CI systems, we must be able to measure and r, decays with exponent δ = 3 [586]. Note that the di- control their dynamic states. 69 5.1.3. Modularity of CIs are unusual events with low probability and catastrophic Modular networks are those with densely connected consequences [608]. It poses challenges for effective groups of nodes while having weak connections be- preparedness for the possibility of CI failures. For ex- tween these groups. CI networks are found to have mod- ample, analysis of the empirical data [609] shows that ular structures. For instance, air transportation networks blackouts have a scale-free distribution for the failure have typical modular structures due to geographical size. That is to say, the possibility of massive outages constraints and geopolitical considerations [597]. The is far beyond the assumption based on the traditional modular structure of a system can be extracted through statistical analysis. It intimates the symptom of self- maximizing the quality function known as “modular- organized criticality [610], where the operating pressure ity” over the possible network partitions [598, 599]. of the system keeps increasing due to the demand for The higher modularity means denser intra-modular con- improving system efficiency. Meanwhile, underlying nections and sparser inter-modular connections. Inter- risks accumulate. These two competing forces drive the modular links are found essential for the system robust- system into a state of criticality. Once perturbation affil- ness, while increasing modularity may lead to decreas- iates into the system, failure size will show a power-law ing robustness against cascading failures [600]. distribution. In the long term, the competition between efficient system operation and reliability improvements 5.1.4. Rich club aspects of CIs is also observed and has generated such unstable bal- Rich club property of complex network suggests the ance [611], which may produce more massive black- dense connections between nodes with high centrali- outs in an unprecedented way. A model example of ties (including degree or betweenness), which forms the the sandpile model [612] can illustrate this idea the in- core structure of the whole complex system. An ana- creasing system pressure (addition of sand) can generate lytical expression is provided to measure and analyze the scale-free distribution of sand cascades. Meanwhile, the rich-club phenomena in different networks, includ- criticality is believed to be different in engineering sys- ing CI [601]. At the autonomous system (AS) level of tems as a result of tuning and optimization. Different Internet topology, the core tiers between nodes show the from many natural networks, CI has significant system rich-club property [602]. For international trade and fi- target, which is usually optimized based on efficiency nancial networks, high-income countries tend to form and cost consideration. This “Highly Optimized Toler- groups, which may spread the financial crisis rapidly ance” (HOT) requires sophisticated topological config- among them [603]. Rich club phenomenon examines uration and can generate high system reliability [613]. the group tendency of prominent elements at the top The HOT model can also produce a power-law distribu- of the network hierarchy to control the majority of a tion of failure size through different mechanisms. system’s resources [604]. Besides the network topol- 5.2.1. Cascading failure model ogy, weighted rich-club networks are a non-trivial gen- eralization, where its metrics and corresponding inte- Given the statistical properties of macroscopic CI grated detection methods are critical [605]. The inter- failures, it is critical to understand and model the mi- national financial system represents a weighted graph, croscopic behaviors. Failure of CI causes substantial where nodes are countries and links are debtor-creditor damages to the system itself and other interdependent relations between countries [606]. Note that the core- systems. When transporting materials and services, CI periphery structure is stable against the 2008 financial faces various types of perturbations, including com- crisis. While performing a critical role in the networks, ponent faults, extreme weather, dramatic changing de- rich-club nodes are also the targets of intentional attacks mands, and malicious attacks. When one or a few [607]. components fail, damages to the failing nodes cause that transports have to go through other routes and they generate extra loads on these routes. These additional 5.2. Failure models of critical infrastructures loads may induce overload on more sites and cause Critical infrastructures usually suffer two significant them to be paralyzed, which is called cascading over- environmental perturbations: extreme weather (e.g., loads [614]. Then, a positive feedback process of over- hurricanes, earthquakes, tsunami) and intentional at- loads is formed and will continue to amplify the dam- tacks (“9.11” attacks, Bali Bombings). Under these per- aging effect. This process does not stop until the sys- turbations, failure of CI has its complexity: at macro- tem reaches a new steady-state, which poses the main scopic, rare events, and at microscopic, cascading fail- threat to the normal operation of critical infrastructures. ures. From the historical statistics, most of CI failures Knowing the propagation behavior of the cascading 70 failure enables us to evaluate the underlying risk and cause partial or complete breakdown of the networks, develop efficient mitigation strategies. Different from with intentional attacks [614, 622] or random perturba- the visible spreading via contacts in most of network tions [616]. In extreme cases, the failure of a single dynamics, cascading failures due to overloads usually node with the largest load is sufficient to destroy the en- propagate through the hidden paths to seemingly un- tire or a substantial part of the system [614]. While the expected locations. Tremendous efforts were made to Motter-Lai model assumes that the overloaded node is isolate or localize the faults based on the assumption permanently removed, Crucitti et al. [44] proposed a that cascading failures are short-range correlated. How- dynamical model with a link efficiency update rules. In ever, the frequency of massive blackouts in the United this model, the overloaded nodes are unsteady but may States are reported not decreasing from 1984 to 2006, recover back functionality, mimicking the congestion despite the enormous investment in system reliability dynamics on the Internet. They also found that attack- [615]. Based on real failure data, jams in city traffic ing a node with the largest load may lower the network or faults in the power grid are spatially long-range cor- efficiency to the collapse level. Simonsen et al. [623] related, decaying slowly with distance [616]. The long- proposed another type of dynamical cascading failure range correlations between failures do not only explain model, and found flow dynamics with transient oscil- why some existing mitigation efforts are inefficient but lations or overshooting may cause more damages than also suggest a new paradigm to tackle this risk. the static models. Load redistribution during cascading Besides the long-range spatial correlation of failures failures is not always deterministic. Thus, Lehmann et in CI, the spatiotemporal spreading of failures is also al. [624] proposed a stochastic model that is analytically unique and worthy of study. One jam in some local solved by generalized branching processes. CASCADE area of a city may cause the subsequent jams nearby or model is developed mainly for cascading model failures locations a few kilometers away. These common phe- on the power grid [621]. This model is at a higher ab- nomena are fundamentally different from the model as- straction level, which could be theoretically solved by sumption of the contagion model in the complex net- the Galton-Watson branching process [625]. work sciences, including SIS and SIR models. Spa- The load redistribution and network structure are es- tiotemporal propagation of overload failures follows a sential for cascading failures. Wang et al. [626] studied wave-like pattern [617], with speed independent of spa- cascading failures on the scale-free networks based on tial network structure. This propagation speed demon- local load redistribution rule. They found that the sys- strates that the failures in CI are not only infectious to tem reaches the most resilient level under specific redis- neighboring sites, but they also spread to sites at a given tribution parameters. Meanwhile, based on the global characteristic distance (measured by constant velocity) load redistribution rule in the Motter-Lai model, Zhao [617]. Confirmed in realistic blackout data analysis, the et al. [627] formulated the cascading process as a phase spreading of failures is non-local with both the topo- transition process. In this model, the cascading failure logical distance and geographical distance [618]. The- causes the full breakdown of the whole network below oretically, this non-local spreading features of cascad- the transition point. Xia et al. [628] studied the cas- ing overloads could be modeled by adding dependency cading failures on the small-world networks, and found links into original network structures [619], as shown that heterogeneous betweenness is one critical factor for in Fig. 72. Although much progress was made thanks network resilience against cascading failures. Wang et to the availability of big data in recent years, predicting al. [629] have studied cascading failures on coupled the outbreak and propagation of cascading failures in map lattices. They found that cascading failures are CI networks is still challenging, yet pressing to become much easier to occur in small-world and scale-free cou- solved. It requires more fundamental understanding and pled map lattices than in globally coupled map lattices. modeling efforts to quantify the cascading failures in CI, While most of the models initiate the cascading over- enabling the design and improvement of system reliabil- loads from given nodes with certain structure features, ity and resilience. percolation framework is also applied to study the criti- A meaningful way to understand the propagation of cal condition of the giant component [616, 630]. cascading failures is based on overload models including the Motter-Lai model [614], the CA models [620] 5.2.2. Critical transitions for transportation and the CASCADE models for power In the past decades, the primary application of the grid [621], etc. In these models, flow dynamics have cascading overload models have been CI networks. For different features, and corresponding cascading failures instance, Wang et al. [631] applied them to the US display specific considerations. The overloads may power grid and surprisingly found that attacking the 71 Figure 72: The effect of a localized attack on a system with dependencies. (A), Propagation of local damage in a system of two interdependent diluted lattices with A B spatially constrained dependency links between the lattices (only one lattice shown here). The hole on the right is above the critical size and spreads throughout the system while the hole on the left is below the critical size and remains essentially the same size. (B), c A localized circular failure of radius rh in a lattice with dependency links of length up to r. Outside the hole, the survival probability of a node increases with the distance ρ from the edge. The parameter ρc denotes the distance from the edge of the hole at which the occupation probability is equal to the percolation threshold of a lattice without dependencies 36 C pc ≈ 0.5927 . (C), Phase diagram of a lattice with D dependencies or two interdependent lattices. Depending on the average degree hki and dependency length r, the system is either stable, unstable or metastable. The circles illustrate the increase (when hki c increases) of the critical attack size (rh) that leads to system collapse in the metastable region. (D), As the system size grows, the minimal number of nodes which cause the system to collapse increases linearly for random attacks but stays constant (≈ 300) for localized attacks. This figure was obtained for a system of interdependent lattices diluted to hki ≈ 2.9 and r = 15 (in the metastable phase-see c), with 1000 runs for each data point. Cited from [619]. lowest load nodes produces more damages than attack- stages. With the consideration of demand and supply ing the highest load nodes. Menck et al. [632] ex- features, including increasing demand and renewable plored the stability mechanism of the power grid, resources, cascading failures could become the first-order vealing that the local structure of dead ends and dead transition in the large size limit [638]. The threat of cas- trees could considerably diminish the system stability. cading failures is also demonstrated in the critical infor- For North American power grids, the failure of a sin- mation structures [639]. This model incorporates two gle substation could lead to up to 25% loss of trans- uncertain conditions regarding the particular next fail- mission efficiency as a result of an overload cascade ure node and time required before the next node state [633]. Besides the US power grid, the Italian power transition takes place. grid [634] and the EU grid [622] were also studied us- Next, we review the critical transitions in transporta- ing the cascading failure models. Blackouts are now tion infrastructure. Based on real-world data, Treiterer mostly modeled by cascading overloads as a series of et al. [640] identified the existence of “phantom traffic outage lines with the consideration of AC or DC flow jams”, where the traffic jams are formed spontaneously dynamics [635]. Except for the short-time scale of cas- with no apparent reason, such as an accident or a bot- cading overloads, the OPA model [636] has been devel- tleneck. Such phantom jams could be traced back to oped to incorporate also the long-term range of power a lane change, or flow dynamics [641]. Furthermore, grid when the network is continuously upgraded to meet large perturbations are found to propagate against the the increasing load demand.The OPA model was val- direction of the vehicle flow [642, 643]. Instead of the idated with data from the Western Electricity Coordi- 1D highway, the formation and propagation of traffic nating Council (WECC) electrical transmission system jams in high-dimensional networks are more compli- [611]. For further understanding of the realistic cas- cated. Equilibrium in traffic is a central concept that cading failures, a simulation model has been proposed determines the flow assignment and resulting distribu- combining both power networks and protection systems tion. In the weighted networks [644], cascading failures [637]. Compared with a simple DC-power-flow based show three types of dynamical behavior: slow, fast, and quasi-steady-state model, this model generates similar stationary. For urban traffic, based on the percolation results for the early stages of cascading as other simula- approach, a new framework for studying urban traffic tors but produces substantially different results for later is proposed [578]. When the roads become congested 72 and are considered “effectively removed”, the emer- addition of these dependency links is found to change gence and formation of urban-scale congestion could the system’s robustness significantly. For the high den- be viewed as a percolation process. The critical point sity of dependency links, interdependent network dis- of this percolation process could be used to evaluate integrates in the form of a first-order phase transition, the resilience of the urban transportation system. Thus, whereas for a low density of dependency links, the sys- the dynamical organization could be analyzed in multi- tem falls apart in a second-order transition [645]. Later scales, showing the spatiotemporal unbalance between on, this framework has been further generalized to han- traffic demand and supply. dle the situation of n interdependent networks , which revealed that percolation in a single network is just a 5.2.3. Interdependent networks limiting case of the general situation of n interdepen- Critical infrastructure systems depend on each other, dent networks [646, 647]. which create more vulnerabilities than a single isolated system does [566]. For example, the 1998 failure of As is the case for CI networks embedded into 2D the Galaxy 4 telecommunication satellite has not only space with cost constraints, once spatially embedded, caused the breakdown of most of the pagers but also the interdependent networks become extremely vulner- disrupted a broad spectrum of other CI systems, includ- able [648]. In contrast to non-embedded networks, there ing financial sectors and emergency systems. The fre- are no critical fraction of dependency links, and any quency of failure propagation between different CI sys- small fraction leads to an abrupt collapse. There could tems is becoming higher due to the increasing embed- be cascading overloads inside each network and depending of the information systems into the existing CI sys- dency failures between different networks [649] with tems to form so-called “Cyber-Physical Systems.” More the consideration of flow dynamics. These combined than that, the interdependence between non-information cascading failures can generate more damages than fail- CI systems is also increasing, given the essential ser- ures in classical interdependent systems. Complex in- vice each CI provideds. For example, blackouts in the terdependence is also modeled as a cyber network over- power grid can cause the breakdown of transportation laying a physical network [650]. An optimum inter-link network or the water distribution network, due to the allocation strategy against random attacks is proposed indispensable role of power for these systems. There when the topology of every single network is unknown. are different types of interdependences in CI, including The interdependent network models enable the design physical dependency, cyber dependency, logical depen- of complex systems with more overall system robust- dency, and so on [566]. This interdependence relation ness and the development of new mitigation methods. can be tight or loose, depending on the functional processes involved. The current models for interdependent CI focus on Despite the existence of models for cascading failures the mapping of hidden functional dependency, differing in interdependent CI, it remains challenging to evalu- from the connectivity links in a single network. In 2010, ate, mitigate cascading failures, and eventually control Buldyrev et al. [14] developed a framework for analyz- the outbreak and propagation of cascading failures. For ing the robustness of interdependent networks subject the state-of-the-art, there are a few measures that can to cascading failures. The failure of even a small num- be adopted to manage the CI failures. Prevention and ber of elements within a single network may trigger a planning are two of the most common methods to avoid catastrophic cascade of failures that destroy global con- catastrophic failures, enabling improvement of the re- nectivity. In a fully interdependent case, each node in silience of CI with resources allocated effectively to var- one network depends on a functioning node in other net- ious parts and stages of CI [651]. Meanwhile, it is dif- works and vice versa. The system shows a first-order ficult to prevent all the CI failures only by planning in discontinuous phase transition, which is dramatically advance and enumerating all the possible failure sce- different from the second-order continuous phase tran- narios. Another standard method is the crisis manage- sition found in isolated networks. This phenomenon is ment [652], which emphasizes the top-down responses caused by the presence of two types of links: connectiv- to the catastrophes. Crisis management must organize ity links within each network; and dependence links be- the mitigation resources timely, especially in the stage tween networks. Connectivity links enable the network of “aftershock”. While these two methods focus on the to carry out its function, and dependence links represent two end stages of cascading failures, a more systematic the fact that the function of a given node in one net- framework for CI management is required. It is the cen- work depends crucially on nodes in other networks. The tral task of system resilience for critical infrastructures. 73 against earthquake threats. Organizational resilience is about the ability of organizations to respond to disaster emergencies while performing the core functions. Social resilience (as discussed in Chapter4) is about the system ability to alleviate negative effects of service losses caused by the social aspect. Economic resilience is to reduce the direct and indirect economic losses as a result of earthquakes. Zobel [121] also proposed sim- Figure 73: Conceptual definition of resilience. ilar resilience definition based on “resilience triangle” Source: Figure from [19]. (Fig. 75), which is then extended to scenarios of partial recovery from multiple disruptive events [661], and defined as 5.3. Infrastructure resilience Resilience concepts have been increasingly applied across a growing number of directions. While reliability [653] emphasizes the system capability of continuously functioning under perturbations, resilience requires the system to bounce back soon and strongly from the serious damages. Resilience is also more than the protection [654], usually referred to the critical infrastructure, and suggests the response and recovery from degradation of system functions. As a new concept for critical infrastructure, resilience has attracted much attention and was defined in different contexts [655, 656, 657, 658, 659, 660]. Resilience emphasizes the adaptation, absorption and recovery from the failure environment. The realization of this comprehensive Figure 74: A framework to assess seismic resilience. ability can be divided into three stages: evaluation, pre- Source: Figure from [19]. diction and adaptation.

5.3.1. Evaluation For critical infrastructure, the ﬁrst step is to evaluate the system resilience. The earlier work to evaluate a system’s resilience was done by the earthquake community, for whom the “resilience” means to measure the system performance under the earthquake and extreme weather. The loss of system performance (single CI or comprehensive urban systems) in the scenario of the earthquake is compared to the benchmark with the consideration of robustness and rapidity [19], as shown in Fig. 73. The loss of resilience can be deﬁned as

Z t1 RL = [1 − Q(t)]dt, (5.1) Figure 75: A definition of resilience. t0 Source: Figure from [121]. RL: resilience loss, Q(t): service quality of the community (limited to the range from 0% to 100%). T ∗ − XT/2 XT R(X, T) = = 1 − , (5.2) T ∗ 2T ∗ It suggests that resilience could be conceptualized in four interrelated dimensions shown in Fig. 74: tech- R(X, T): predicted resilience function, nical, organizational, social, and economic. Technical X: the initial loss value, resilience is about the performance of physical systems T: the recovery time, 74 T ∗: a strict upper bound on the set of possible sessment. Chang et al. [664] have defined a probabilis- values for T. tic resilience based on the probability of the initial system performance loss after a disruption being less than Mathematical models for evaluation. Henry et al. the maximum acceptable performance loss and the time [662] developed a time-dependent resilience metric to to full recovery being less than the maximum accept- measure the system resistance to disruption during dif- able disruption time. This is different from determinis- ferent stages including reliability, vulnerability and re- tic metric, which is mainly for disasters with complete coverability, which is written as information, and can be written as φ(t |e j) − φ(t |e j) | j r d ∗ ∗ Rφ(tr e ) = j , (5.3) R = P(A|i) = P(r0 < r and t1 < t ), (5.6) φ(t0) − φ(td|e )

j R: system resilience, Rφ(tr|e ): the value of resilience, indicates the proportion of delivery function that has been P(A|i): (resilience defined as) the probabil- recovered from its disrupted state, ity that the system of interest will meet pre- φ(): delivery function (or the so-called figure- defined performance standards A in a scenario of-merit), seismic event of magnitude i, e j: disruptive event, r0: the initial loss, r∗: maximum acceptable loss, tr: the set time for evaluating the current system resilience, t1: the time to full recovery, t∗: maximum acceptable disruption time. td: the time when the system transits to its final disrupted state, Ouyang et al. [665] have proposed a resilience metric t : original time. 0 based on the expectation of system performance under Francis et al. [663] have proposed a dynamical re- perturbations, which is silience metric, which explicitly incorporates the recovery speed. They also consider two dimensions of a sys-  R T   P(t)dt  tem capable of absorbing perturbations and adaptive ca- AR = E  0  R T  pacity to recover in the post-disaster stage, which is as  TP(t)dt  follows 0 (5.7) R T PN(T)   TP(t)dt − AIAn(tn) = E  0 n=1  , Fr Fd  R T  ρi = S p , (5.4)  TP(t)dt  Fo Fo 0   ∗ ∗ ∗ AR: a time-dependent expected annual re- (tδ/tr )exp[−a(tr − tr )] for tr >= tr , S p =  ∗ (5.5) silience metric, (tδ/t ) otherwise, r E[]: the expected value, ρi: resilience factor, T: a time interval for a year (T=1 year=365 S p: speed recovery factor, days), Fr: the performance at a new stable level after P(t): the actual performance curve, recovery efforts have been exhausted, TP(t): the target performance curve, Fd: the performance level immediately post- N(t): the total number of event occurrences disruption, during T, Fo: the original stable system performance tn: the occurrence time of the nth event, level, AIAn(tn): the area between the real perfor- tδ: slack time, mance curve and the targeted performance tr: time to final recovery (i.e., new equilib- curve (i.e., impact area), for the nth event oc- rium state), currence at time tn. ∗ tr : time to complete initial recovery actions, a: parameter controlling decay in resilience Youn et al. [666] have considered the resilience as the attributable to time to new equilibrium. sum of the passive survival rate (reliability) and proactive survival rate (restoration), different from the most of Besides the deterministic definition of resilience, prob- definitions that integrate the system performance during abilistic metric is also considered in the resilience as- the whole disruption process, which is defined as 75 silience index. Comes et al. [671] evaluate the resilience of power grid and New York subway systems, Φ(resilience) = R(reliability) + ρ(restoration), (5.8) and compares them with such infrastructure in different regions. Φ(resilience): the conceptual definition of engineering resilience, For the resilience of transportation networks, Ip et R(reliability): the degree of a passive survival al. [672] have proposed a network resilience met- rate, ric expressed as the weighted sum of node resilience, ρ(restoration): the degree of a proactive sur- which for each node is evaluated by the weighted aver- vival rate. age number of reliable passageways to other nodes in the network. With this resilience metric, transportation Ayyub [667] has incorporated the effect of system aging networks are optimized with a structure optimization in the resilience assessment, including different failure model. For information CI networks, resilience is also profile of brittle, ductile, and graceful failures, which defined accordingly but it also includes concepts devel- follows the following definition oped earlier from survivability [673, 674]. Sterbenz et al. [673] have defined resilience at any given layer as a Ti + F∆T f + R∆Tr (negative) change in service corresponding to a (nega- Re = , (5.9) Ti + ∆T f + ∆Tr tive) change in the operating conditions. Network operational space is divided into normal operation, partially Re: system resilience, degraded, and severely degraded regions, while the ser- Ti: time to incident, F: failure profile, measuring the system’s ro- vice space is divided into acceptable, impaired, and un- bustness and redundancy, acceptable regions. Then, the resilience is represented R: recovery profile, measuring the system’s for a particular scenario at a particular layer boundary resourcefulness and rapidity, as the area under the resilience trajectory, as shown in Fig. 76. Fang et al. [675] have proposed two metrics of ∆T f : failure duration, optimal repair time and resilience reduction metric, to ∆Tr: recovery duration. evaluate the component importance. These two metrics Conceptual models for evaluation. Ouyang et al. can determine the repairing priority of failed component [665] has proposed a multi-stage framework to analyze to gain higher resilience, and the potential losses if the infrastructure resilience. These three stages are divided component repair is delayed. as: disaster prevention from normal operation stage, cascading failures from the initial failure stage, and the recover stage. Taking the power transmission grid in Harris County, Texas, USA, as a case study, the authors compare an original power grid model with several resilience models to validate the effectiveness of these models under random hazards and hurricane hazards. Cai et al. [668] have proposed an availability- based engineering metric to measure resilience, which depends mainly on engineering system structure and maintenance resources. System resilience is evaluated by the developed method based on dynamic-Bayesian- network. Renschler et al. [669] have defined resilience in seven dimensions (so-called PEOPLES, P: Popula- tion and demographics; E: Environmental/Ecosystem; O: Organized governmental services; P: Physical infrastructure; L: Lifestyle and community competence; Eco: Economic development; and S: Social-cultural capital). Based on Renschler’s definition, Vincenzo et al. [670] have analyzed the transportation system resilience under extreme events. Considering the interdependencies between systems, categories and dimensions, any re- Figure 76: Resilience measured in state space. covery plan could be evaluated by maximizing the re- Source: Figure from [673]. 76 Spatio-temporal network models for evaluation. Un- lihood of the successful system functioning during its til now, most of the resilience metrics are dimensionless, whole lifetime. In other words, reliability engineering describing the system performance variation during the requires inherently proactive management of whole sys- perturbations. However, critical infrastructures are typ- tem lifetime, including design, testing, operation, main- ically spatial-temporal systems, failure of which spans tenance, and so on [684]. Reliability engineering is first along the space and time. This spatial-temporal failure applied to electronic equipment because of its high fail- behavior and corresponding system response requires ure rate, while reliability engineering for complex sys- more sophisticated framework to evaluate and analyze tems is now different. For example, fault tree analy- the system resilience. In [676], Zhang et al. proposed sis, a classical and efficient assessment for system re- a spatio-temporal definition to reflect the resilience fea- liability, could help to locate the root cause for system ture of CI systems, as shown in Fig. 77, which is defined failure. For complex systems, the emergency of system as failures is usually not the sum of independent compo- Z t 1 nent failures, which poses challenges for fault tree anal- S = Ms(t)dt, (5.10) t0 ysis and other traditional reliability analysis methods. S : spatio-temporal resilience loss, Actually, classical reliability methods assume the absence or weak coupling between components, protocols Ms(t): size of the failure cluster at a snapshot of the temporal layer t. or failures. One possible solution is to combine the reliability engineering with network science [685], which This resilience quantification integrates the spatial is proved valid when analyzing complex systems [686]. spreading of traffic congestions during its lifespan. For network systems, existing reliability studies focus More surprisingly, with the new spatio-temporal defi- on connectivity. Connectivity reliability measures the nition of system resilience, it is found that the system probability of the existence of connection paths between resilience follows a stable scale-free distribution, which network nodes. Connectivity reliability can be further is usually the result of self-organized criticality. This re- classified as two-terminal, k-terminal and all-terminal sult contribute to the longstanding discussion of whether problems [687, 688, 689]. The network is considered resilience is an intrinsic property of system [663], by structurally reliable if there is at least one path between finding that traffic resilience has similar scaling laws for required terminals. Considering the redundancy, natu- different days and different cities. This scale-free distri- ral connectivity is proposed with acute discrimination bution is stable across different working days in Beijing for different networks and derived from graph spectrum and Shenzhen cities and is found to depend only on a [690]. Besides the reliability metric based on probabil- few macroscopic parameters, including system dimen- ity theory, another metric is proposed and referred to as sion and demand. Besides the system resilience, the re- belief reliability [691], which can measure the aleatory covery time of jamming failures is also found to follow uncertainty and also epistemic uncertainty. Connectiv- a scale-free distribution, but with a different exponent. ity reliability is usually considered in the scenarios of Note that, on the temporal scale, the scaling relation be- earthquakes [692], flood [581, 693], or other emergency tween the lifetime of the traffic jam and system size is situations [694], where the state of roads are binary found in a 1D lattice cellular automaton model [677]. without considering the travel dynamics. Meanwhile, On the spatial scale, spatial correlations in traffic flow besides the basic connectivity function, reliability also fluctuations are also found to follow a power-law de- refers to a certain service level from the user demand cay [616, 678]. With these scaling on the spatial and side. Travel time reliability is then developed to fill this temporal scales, a combined spatio-temporal scaling of gap [695, 696], which is the probability that a trip could traffic jams is found, viewing jams in a stereo way. Re- be completed within a given time duration. Asakura silience independent of typical efficiencies [679] may be [697] has studied reliability measures between an origin explained by the stable resilience scaling. and destination (OD) pair in a degraded road network Reliability of networks. System reliability [680] refers with links damaged by natural disasters. While these to fundamental quantities for system resilience. Reli- types of reliability index have their own advantage, it ability engineering focuses on the ability of a system remains challenging to measure the reliability in a more or component to function under stated conditions for a comprehensive way. Chen et al. [698] have made an at- specified period [681]. Besides the statistical measure- tempt to define travel capacity reliability. It is defined as ment of failures, reliability engineering pays more at- the probability that a network can meet a certain level of tention to the analysis of failure mechanisms [682, 683], travel demand at the required service level. Considering prediction of possible causes, and improvement of like- the demand fluctuation, Shao et al. [699] studied traffic 77 Figure 77: Traffic resilience defined based on spatio-temporal jammed clusters. (A) Illustration of the evolution of a jammed cluster in a city. Red links are considered congested. All red links in the shadow belong to the same jammed cluster. (B) The cross-section area Ms(t) of the second largest jammed cluster on October 26, 2015 in Beijing. Since the resilience is reduced during the jam, we plot the negative of Ms(t) as a function of time, and traffic resilience can be represented by the gray area. The gray area is the size of the spatiotemporal jammed cluster (S) shown in red in A. The time span between t0 and t1 represents its recovery time (T = t1 − t0 + 1). (C) The cluster sizes of the first, second, and third largest jammed clusters on October 26, 2015 in Beijing as a function of time (the second and third largest clusters sizes are given on right-axis scale). Cited from [676]. reliability with a stochastic user equilibrium traffic as- of the system, which can be measured by the severity signment model, which is solved by a proposed heuris- of harmful consequences and its probability [705, 706]. tic solution algorithm. To explore the spatio-temporal The other approach considers the risk from the Bayesian dimension of travel reliability, Li et al. [578] have pro- perspective [707], and focuses on development of an posed a new definition to measure the operational relia- epistemic framework. Overall, risks come from the in- bility of urban traffic with the dynamical traffic cluster, herent uncertainties in the system itself and our limited which is only composed of high-velocity roads. This knowledge about the system. These uncertainties of un- can describe how well the network traffic above a cer- expected and adverse consequences could pose various tain service level covers the city, from the viewpoint of threats to the systems. While these uncertainties in the network operators instead of a single user. risk assessment can hardly be fully realized, resilience may help to provide a complementary method for risk Since the introduction of the resilience concept into management, which builds the system’s ability to ab- the study of critical infrastructure, resilience engineer- sorb or adapt to these uncertainties. ing is related to safety management [700], which is also the focus of risk analysis. Given the unprecedented risks Vulnerability in networks. Besides uncertainty evalu- and possibly catastrophic damages resulting from them, ation in the risk analysis, another critical step is to ex- risk analysis [701] is also one of the critical methods plore the vulnerable part of the system, which is the to understand the system resilience. Critical infrastruc- mitigation focus on the system recovery. Vulnerabil- tures are usually threatened by natural hazards, where ity is usually defined through the degree of exposure risk assessment is related to the vulnerabilities exposed to hazards and the degree of losses due to the risks to these disasters [702]. Hazard assessment mainly [708, 709, 710, 711]. For CI networks, another kind of studies the disaster itself, while vulnerability focuses on features, spatial properties [648], are recognized now to the exposure and economic losses of critical infrastruc- be another source of vulnerability. Different indicators tures caused by failures. Interdependence is also an- can measure the vulnerability of a link as the service other source for system vulnerability [703]. Concepts degradation due to to the possibility of it being closed. of uncertainties in risk usually have two angles [704]. For road networks, several link importance indices and The first type considers risk as the inherent properties site exposure indices are proposed [712], based on the 78 variation in generalized travel cost when these links cupancy tenure to provide a predictor of business loss. are disrupted. For power grids, based on network- Hong et al. [730] have performed vulnerability analysis survivability analysis, identifying the most vulnerable for the Chinese railway system under the flood risks and locations in the power grid is studied [713], considering provided an effective maintenance strategy considering the spatial correlation between outages. A water distri- link vulnerability and burden. bution network is critical for city life and therefore may Critical infrastructures are usually modeled as spa- be the target of intentional attacks. The vulnerability of tially embedded networks, where vulnerability analysis the water distribution network is found to depend on hy- has also found interesting and surprising results. The draulic analysis and network topology [714]. Chang et natural disaster, including earthquakes and landslides, al. [715] have proposed measurement for post-disaster could be modeled by localized damages or attacks to transportation performance. Tuncel et al. [716] have the spatial network of critical infrastructures. Berezin proposed a risk management for supply chain networks et al. [619] have proposed a general theoretical model based on failure mode, effects, and criticality analysis for localized damage to a spatially embedded network (FMECA) technique and Petri net simulation. with functional dependency. They have found that lo- In some cases, critical infrastructures are subject to calized damage can cause substantially more harm to multiple hazards. While some related studies [717, spatial networks than equivalent random damage. Lo- 718, 719] have focused on analyzing the probability calized damage will generate cascading failure when of joint hazard occurrences, corresponding risk assess- the attack size is larger than a critical value indepen- ments method for this scenario have also been pro- dent of the system size (i.e., a zero fraction). Empirical posed [720, 721]. For example, considering multiple study [618] also confirms this modeling result. With the hazards of earthquake and hurricane, Kameshwar et data of the North American power grid from 2008 to al. [722] have proposed parameterized fragility based 2013, they found that significant cascading failures are multi-hazard risk assessment procedure for highway usually associated with concurrent events closer to each bridges. other and the vulnerable set. Localized damage models Vulnerability in transportation is defined [723] as “a have also been extended to interdependent infrastruc- susceptibility to incidents that can result in consider- tures with terrorist attacks [731, 622] or vulnerability able reductions in road network serviceability”. Com- analysis [732]. pared with reliability, vulnerability usually refers to more harmful consequences but lower probability to oc- 5.3.2. Prediction cur [724]. Based on an integrated equilibrium model Prediction of the spatio-temporal propagation of cas- for a large scale transportation system, Nicholson et al. cading failures could determine the timing and amount have studied the critical components based on its socio- of mitigation resource allocation in corresponding real- economic impacts [725]. time resilience management. Meanwhile, uncertainties In the network analysis, vulnerability is related to in the emergence and proliferation of cascading over- nodes or links with specific topological features. For loads bring fundamental unpredictability to some ex- example, the vulnerable part of a scale-free network is tent. For CI and other engineering systems, resilience usually considered to have the largest degree (i.e., num- studies mostly assume a single equilibrium and focus ber of links) [726]. Other topological features are also on the system’s ability to “bounce back” to the original found to be related to the vulnerability. Weak rela- state after perturbations. As the dynamical system com- tionships between different communities in a modular plexity demonstrated, systems in practice could have network can do extensive damage when it is removed. multiple states, which could be studied by the theory Critical links in the airline network are explored with of multi-state system [733, 734, 735, 736]. Multi-state memetic algorithms [727], and are found different from systems are now applied to evaluate and optimize the those with the highest topological importance. Basoz system reliability [737, 738, 739]. While the multi-state [728] has proposed a risk assessment method consisting system requires the availability of probability data for of three parts: (i) retrofitting of critical infrastructures all the system components, it is rarely possible in the as a means of pre-disaster mitigation, (ii) pre-disaster real case due to the limited budget and time for observa- emergency response planning, and (iii) emergency re- tion. Thus, the theory of the fuzzy set is combined with sponse operations. He has measured the importance of the traditional multi-state system to gain a realistic esti- components by network analysis and decision analysis. mation with an acceptable computation cost. Ding et al. Chang et al. [729] have proposed a vulnerability met- have extended the multi-state system analysis through ric based on the business sector, size, and building oc- the development of a fuzzy universal generating func- 79 tion [740]. In this analysis, performance rates and cor- Systems may undergo a regime shift of abrupt transi- responding probabilities are handled with fuzzy values. tion when evolving from one metastable state to another. A general model for the fuzzy multi-state system is also This regime shift is usually highly unpredictable, which proposed [741]. poses a significant challenge for the prediction in com- For ecological or climate systems, there exist multiplex systems [181]. The tipping point often marks the ple equilibrium states, allowing the system to shift from critical threshold of a regime shift. When a system is ap- one to another state under a given disturbance at the tip- proaching its tipping point, a positive feedback mecha- ping point [742]. Critical infrastructure states can also nism pushing system to another state is becoming much evolve into different states in daily operations or emer- more substantial. It leads to a domino effect of cascad- gencies. Moving back to the original state may require ing failures when the robustness or reliability of criti- reaching a different tipping point, it is called hystere- cal infrastructures is concerned. Homogeneous systems sis property [743], which is widely observed in vari- with dense connections may have strong local resilience ous physical systems. Among other critical infrastruc- yet be fragile globally [181]. tures, the hysteresis of traffic dynamics is mostly stud- Meanwhile, heterogeneous modular systems could ied. Traffic hysteresis in freeway traffic was firstly ob- have the adaptive capacity with gradual changes to served in the real data in 1974 [640]. A traffic jam can environmental variations. The relation between sys- spontaneously appear during phase transition without tem resilience and structure can help to predict social- obvious reasons in free traffic flow, accompanied by a technological infrastructures better. Prediction of close- hysteresis phenomenon [744]. Fundamental relations in ness to the tipping point is an urgent and critical need for traffic between velocity, flow, and density or occupancy avoiding catastrophic collapse and improving system re- have been observed for a long time [745, 746]. The silience. One classical work is “critical slowing down”, so-called fundamental diagram is widely accepted and suggesting that the rate of system recovery to the orig- identified in different scenarios, e.g., the flow-density inal state is becoming lower when it is approaching the relation. The fundamental diagram could be derived in vicinity of the tipping point [213, 380, 215, 222, 216, different ways [747, 748]: (1) from statistical model- 217]. When systems have intense fluctuation, an altering; (2) from car-following, and (3) from fluid analo- native indicator is developed to infer the shape of system gies. However, this diagram has been challenged by state landscape [754, 755]. The change of potential sys- traffic hysteresis: for example, the relationship between tem landscape also reflects the shifting between differ- density and speed is found to show a “loop” behavior. ent attractors. While these indicators have been success- Different theoretical models are proposed to explain fully apply in different disciplines, a development of a the phenomenon of hysteresis. It is conjectured [750] valid applicable indicator is far from completion. As that hysteresis is generated by the asymmetrical driver the availability of high-resolution spatio-temporal data behavior between acceleration and deceleration. Zhang is increasing, more indicators could be explored, includ- [748] has proposed a mathematical theory to model the ing spatial information, which is sometimes more infor- hysteresis phenomenon, where acceleration, decelera- mative and robust than temporal indicators, especially tion, and equilibrium flow is distinguished. Empirical for critical infrastructures. results are well predicted by the proposed approach. Chen et al. [751] have also suggested that traffic hys- 5.3.3. Adaptation and control teresis will occur when drivers’ reaction to traffic os- The ultimate goal of system resilience is to improve cillations is not symmetric. With the incorporation of the system’s ability to absorb and adapt to unexpected velocity-dependent randomization, Barlovic et al. [752] risks and faults. With the development of big data and have demonstrated the metastable states with the Nagel- control technology, CI systems have become more and Schreckenberg model through the slow-to-start behav- more automated and intelligent. Meanwhile, one sig- ior. Rather than 1D lattice used in most of the above nificant difference of CI systems compared with other models, Hu et al. [753] have found hysteresis phe- engineering systems, is that human behaviors and de- nomena of information traffic on the scale-free network. cisions are deeply involved in every inch of CI sys- They have observed hysteresis loop of two branches tems. For example, traffic congestion has become an for the fundamental diagram: the upper branch is ob- “urban disease” that impedes urban development. The tained by adding packets in the free-flow state, while the primary “pathogenesis” may be due to the imbalance lower branch is obtained by removing packages from between supply and demand of spatio-temporal trans- the jammed state. These model results suggest the exis- portation resources, which is manifested by the mis- tence of multiple states of traffic dynamics. match between the rapid growth of short-term traffic 80 A B C

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Figure 78: Bring a network to its desired state through perturbation of the initial state. (A) The goal is to drive the network to a desired state by perturbing nodes in a control set: a set consisting of one or more nodes accessible to compensatory perturbations. (B) State space portrait. In the absence of control, the network at an initial state x0 evolves to an undesirable equilibrium xu in the n-dimensional state space (red curve). By perturbing the initial state (orange arrow), the network reaches a new state that evolves to the desired target state x∗ (blue curve). (C) In this example, the network is controllable if and only if the corresponding slice of the target’s basin of attraction (blue volume) intersects this region of eligible perturbations (grey volume). (D) A perturbation to a given initial condition (magenta arrow) results in a perturbation of its orbit (green arrow) at the point of closest approach to the target. (E) This process generates orbits that are increasingly closer to the target (dashed curves), and 0 is repeated until a perturbed state x0 is identified that evolves to the target. Source: Figure from [749]. demand and the slow improvement of traffic supply ca- 757, 759, 760]. For example, connections could be pacity. The urban traffic has the following features: in- classified into three groups (critical/redundant/ordinary) creasing in the short-term, being distributed differently based on their influence on the controllability of the net- in the whole space, and jamming quickly in the local works [756]. area. Under the circumstance of limited transportation From the system control view, the realization of sys- resources, the congestion can become challenging to tem resilience for recovering to a healthy state depends dissipate and may propagate over space and time, result- on if we could tune the system to the desired attractor in ing in possible regional dysfunction of urban traffic. In terms of control theory. For linear systems, this could extreme weather, important events, or other emergency be achieved by minimal control energy strategy. How- situations, city-scale congestion and significant degra- ever, most of the critical infrastructures are not linear dation of traffic capacity may happen, which calls for systems. For control of nonlinear systems, especially the system adaptation and recovery capability. those with numerous components and complex interac- Adaptation and recovery of CI could be achieved tions, currently there are no general solutions. Mean- by the ultimate control of complex systems. Recently, while, different approaches have been proposed consid- the study of controllability attracts much attention to- ering the specific background of problems. For urban wards the control of complex networks [756, 757, 758]. traffic, the system is found to follow the dynamic pat- These studies offer theoretical tools to identify a sub- tern of the macroscopic fundamental diagram (MFD), set of driver nodes to control the whole systems through which allows the controller to perform state control ac- proper control strategies. Liu et al. [756] have devel- cordingly. For example, optimal perimeter control for oped conceptual tools to study the controllability of an two-regions is proposed through model predictive con- arbitrary complex directed network, through the set of trol [761], with the controller at the border between two- driver nodes with time-dependent control for the sys- regions. For the power grid, Cornelius et al. have pro- tem’s entire dynamics. With this framework of network posed to bring a network to its desired state through per- controllability, nodes and links could be ranked based turbation of the initial state to the attractor basin of de- on its importance in controlling the whole network [756, sired state [749], as shown in Fig. 78. 81 For critical infrastructure, a self-healing framework has been proposed to realize the system adaptation [564], as described in Fig. 79. This framework is based on intelligent agents distributed over the whole system at its different levels. These agents will make decisions according to their environments and interact with each other. Liu et al. [762] have proposed a Knowledge/decision exchange self-healing model against cascading overloads, where Hidden failure Reconfiguration monitoring agents agents could decide to restore timing and resources. agents An optimal restoration timing is found to recover the Vulnerability system from the edge of collapse. Lin et al. [763] assessment Restoration agents agents have proposed a self-healing transmission network reconfiguration algorithm, with the consideration of elec- Event Planning identification agent trical betweenness. Quattrociocchi et al. [764] realized agents the self-healing capability with distributed communication protocols. They have studied the effects of re- Triggering events Plans/decision dundancy under different connectivity patterns on heal- Event/alarm Model Command filtering update interpretation ing performances. Gallos et al. [765] have proposed a agents Update agents Check agents model consistency local-information based self-healing algorithm through

Events/alarms controls adding after damage links as short as possible, considering the fraction of failed neighbors. Shang [766] studied Fault Frequency the impact of self-healing capability on network robust- isolation stability agents agents ness, with the percolation framework. Inputs Inhibition signal Protection Protection agents agents 6. Resilience, robustness, and stability

Power system Two concepts “robustness” and “stability” are quite related to “resilience”. They are commonly used to analyze the system response under changing conditions, and sometimes difficult to distinguish due to the lack of clear boundaries [767], and their own definitions may vary across different contexts. For example, in the study Figure 79: A multi agent system design. This design organizes of how a system responds to changes in a specific driver, agents in three layers: The reactive layer (bottom) consists of agents Dai et al. [768] define resilience as the size of the that perform reprogrammed self-healing actions that require an basin of attraction and stability as the recovery rate. immediate response. Reactive agents, whose goal is autonomous, fast control, are in every local subsystem. The agents in the coordination Whereas in the bacterial response to antibiotic treatment layer (middle) include heuristic knowledge to identify which [390], resistance describes the insensitivity to the treat- triggering event from the reactive layer is urgent, important, or ment and resilience defines the recovery rate. Especially resource consuming. These agents, whose goal is consistency, also update the system’s real-world model and check if the plans (or resilience and robustness, they have been used as in- commands) from the deliberative layer (top) represent the system’s terchangeable concepts in some literatures on network current status. If the plans don’t match the real-world model, the analysis [1, 769]. Here we trace back to their original lu- agents in the middle layer trigger the deliberative layer to modify the cid definitions on resilience and stability that were first plans. The deliberative layer consists of cognitive agents that have goals and explicit plans that let them achieve their goals. The goals proposed in ecology [4], and the robustness in complex of agents in this layer are dependability, robustness, and self-healing. networks [770]. We point out that these three notions Source: Figure from [564]. describe distinct properties of systems. Resilience determines the tolerance of a large perturbation without shifting to an alternative stable state, being a measure of the ability of these systems to absorb changes of state variables, driving variables, and parameters, and still persist [135]. In this definition, resilience is the property of the system with persistence or probability of extinction being the result. Whereas, stability is 82 the ability of a system to return to a specific stable fixed and engineering resilience is the speed of return to the point after a temporary disturbance. The more rapidly steady state held before the disturbance. Adaptive re- it returns, and with the least fluctuation, the more sta- silience is the capacity to remain productive and true to ble it is. In this definition stability is the property of the core purpose by adapting and transforming to chang- system with the degree of fluctuation around a specific ing circumstances [776, 23]. As we reviewed in the state its measure [4]. Resilience and stability are both previous four sections, these three types of resilience defined on network dynamics [4, 12], while the concept have not only been applied to low-dimensional dynam- “robustness” is related to the static structure of a net- ical systems [28] to predict their response to changes, work, measuring the ability to maintain its connectivity but also to high-dimensional networks [12]. In complex when a fraction of nodes (links) is damaged [40], Next, networks, network dynamics are the underlying mecha- we give further clarification on the differences between nisms required for resilience analysis. Next, we review them. more studies on network dynamics. Network dynamics. Network dynamics describes how 6.1. Resilience and network dynamics the entities evolve over time, for example, how the gene The word resilience is derived from the Latin terms expression level changes in gene regulatory networks resiliere or resilio for ‘bounce’ or ‘rebound’ [771]. The [777]; how the specie abundance evolves during a time action of “bouncing back” characterises the basic mean- period [778]; how many individuals are newly affected ing of resilience, which is a dynamical property that re- by or recovered [779] from a certain disease? Modelling quires a shift in the system’s core activities. In other such processes require determinations of equation forms words, network resilience is a concept describing net- and a set of parameters, which are not easy to achieve works’ ability to retain its basic functions defined on due to the complexity and unknown mechanisms within network dynamics within a certain region and there is these processes. Thus, comparing to extensive stud- no requirement to stay in a specific fixed point, which ies on reconstructing networks’ static structures [780], is different from the concept of stability. Resilience was there are much less research on modelling large scale originally used in material science to describe the resis- networks’ dynamics. In the following, we review the tance of materials to physical shocks [772], and widely dynamical models of several large scale systems. used to characterize individual’s ability to cope with ad- Systems may exhibit various forms of nonlinear dy- versity, trauma or other sources of stress [773, 767]. namics [781], such as oscillation, spreading and bifur- In 1973, Holling et al [4] define resilience as a mea- cation. The dynamic models of them involve different sure of persistence of systems and their ability to ab- equation forms and different parameters. For exam- sorb change and disturbance, which has become a pop- ple, the networked Stuart-Landau (SL) oscillators sys- ular concept being widely used in ecology. Later on, tem [782, 783] can be described by the following cou- resilience has been adopted as a generic concept to de- pled ordinary differential equations (ODEs): scribe the response to changes in systems from other N fields, such as biology [371], social science [518], en- dz j X = (α + iΩ − |z |2)z + σ c (z − z ), (6.1) gineering [569], etc. For most of these applications, re- dt j j j j k j k j k=1 sistance to perturbations and rate of recovery after their occurrence are considered the key aspects of resilience where z j and Ω j are the complex amplitude and the in- [21]. Based on the adaptive cycle that includes four herent frequency of the jth SL oscillator, and α j is its phases: growth, consolidation, release, and reorganisa- control parameter. Parameters σ and ci j describes the tion [774, 775], a more general meaning of resilience interactions between the jth and other SL oscillators. was proposed [24, 17], which is the capacity of social- Figure 80 shows the (a) active and (b) inactive dynam- ecological systems to adapt and transform in response ics in an isolated oscillator. to unfamiliar, unexpected, and extreme shocks. The dynamics in gene regulatory networks can be Resilience is probably the broadest concept among modelled as Michaelis-Menten equations [784, 785] the three concepts discussed in this section [767], and N xh there are three types of resilience: ecological resilience dxi X j = −Bxa + A , (6.2) (or termed as system resilience) [4], engineering re- dt i i j h j=1 1 + x j silience [21] and adaptive resilience [774, 24]. As discussed in Chapter.2, ecological resilience is defined where B is the rate constant, a is the level of self- as the magnitude of disturbance that can be absorbed regulation and the Hill coefficient h describes the level before the system shifts to an alternative stable state, of interactions between genes. The spreading process 83 A B Recently, Barzel et al. [52] develop a method to 1.5 1.5

1 1 infer the microscopic dynamics of a complex system

0.5 0.5

) ) from observations of its response to external perturba- j j 0 0 tions, enabling the constructions of nonlinear pairwise Re(z -0.5 Re(z -0.5

-1 -1 dynamics that are guaranteed to recover the observed -1.5 -1.5 behaviour. Given a complex networked system with N 0 5 10 15 0 5 10 15 components, whose state x (t), (i = 1, ...N) are govern Time Time i by the following ODEs Figure 80: The (a) active and (b) inactive dynamics in isolated oscillators. Periodic oscillation presents in the active oscillator while XN inactive oscillator becomes quiescent after transient damping dxi = M0(xi(t)) + Ai j M(xi(t), x j(t)), (6.5) oscillation. Here, we set the control parameter α = 2 for the active dt j j=1 oscillator and α j = −2 for the inactive oscillator. The inherent frequency Ω j is 2. where M0(xi(t)) describe the ith component’s self- dynamics, and M(xi(t), x j(t)) captures the impact of can be modelled as a susceptible infected susceptible neighbour j on the state of i. Ai j is the adjacency ma- (SIS) model [786] trix. Factoring the interaction term as M(xi(t), x j(t)) = M1(xi(t))M2(x j(t)), the system’s dynamics is uniquely dx XN characterised by three independent equations i = −Bx + A R(1 − x )x . (6.3) dt i i j i j j=1 m = (M0(x), M1(x), M2(x)), (6.6)

All these dynamic equations can be generalised by the which is a point in the model space M. For systems of following equation unknown microscopic dynamics, the challenge is to infer the appropriate model by identifying M0(x), M1(x), N dx X and M (x) that accurately describe the system’s observ- i = W(x (t)) + A Q(x , x ), (6.4) 2 dt i i j i j able behaviour X. Define a subspace (X) ∈ com- j=1 M M prising all models m that can be validated against X. where the first term W(xi(t)) describes the self- Barzel et al. [52] proposed a method to link the ob- dynamics of xi, and the second term captures the in- served system response to the leading terms of m. It de- teractions between the component i and its neighbours. fines the exact boundaries of M(X) rather than a specific Barzel et al. [242] develop a self-consistent theory of model m, providing the most general class of dynamics dynamical perturbations to uncover the universal char- that can be used to describe the observed responses cap- acteristics in a broad range of dynamical processes. By tured by two quantities: the transient response T and analysing the macroscopically accessible distributions the asymptotic response G. By applying to both numer- of the following dynamical measures: the correlation ical data of gene regulatory dynamic model and empiri- G, impact I, stability S , the propagation and cascade cal data of cell biology and human activity, the effective dynamics, the system’s universality class could be de- dynamic model can predict the system’s behaviour and termined even without knowledge of the analytical for- provide crucial insights into its inner workings. mulation of the dynamics within the system. In summary, the prediction of resilience in large scale Despite the difficulties in uncovering the complex and networks have been restricted by the analytical tools nonlinear mechanisms of network dynamics, the accu- and the unclear internal network dynamics for a long mulations of massive data and rapid development of time. With the developed general frameworks, such as computational methods make it possible to identify and the dimension-deduction method proposed in Ref. [12], predict dynamic models directly from empirical data. and the methods for inferring network dynamics as we Takens [787] proved that the underlying dynamical sys- reviewed above, more and more studies could be car- tem can be faithfully reconstructed from time series un- ried out to predict the resilience in complex networks in der fairly general conditions, establishing a one-to-one future. correspondence between the reconstructed and the true but unknown dynamical systems [49]. Wang et al. [50] 6.2. Stability in complex systems and Yang et al. [51] develop a framework based on com- The word “stability” is derived from the Latin term pressive sensing, which is able to predict the exact forms stabilis, which means firm or steadfast. In a dynamical of both system equations and parameter functions. system, stability defines the system’s ability to stay in 84 an equilibrium state. Stability has a rich history in ecol- eigenvalues of the adjacency matrix of the networked ogy [788]. There, stability was first defined as the con- system [799, 800]. The largest real part of the eigenval- stancy of a given attribute, regardless of the presence ues determines whether, and how fast, the system will of disturbing factors [789, 790]. For instance, stable return after perturbation[801]. If this quantity is neg- ecological communities were those with relatively con- ative, the system is stable; more negative values indi- stant population sizes and compositions [791]. Later, cate that the system returns to stability more quickly. the definition of stability has been expanded to de- The imaginary parts of the eigenvalues predict the ex- scribe other properties of ecosystems, such as the abil- tent of oscillations in species densities during a return ity to maintain ecological functions despite disturbances to equilibrium: Larger imaginary components predict [792], or the ability to return to the initial equilibrium more frequent oscillations [802]. Allesina et al. [801] state [4, 788, 793]. This led to multiple definitions and proposed analytic stability criteria for complex ecosys- interpretations of stability, and overlaps with the con- tems including different kinds of interactions: preda- cept of resilience [794, 795, 796, 767]. For example, tor–prey, mutualistic and competitive. Even the stability for systems with alternative stable states, one concept criteria are proposed especially for ecosystems, they are of stability depends on the number of alternative stable widely applicable, because they hold for any system of states: more stable systems are those with fewer sta- differential equations [801]. ble states [788]. Another concept of stability describes the ease with which systems can switch between alternative stable states, with more stable systems having higher barriers to switching [788]. The latter concept is quite the same as the meaning of resilience. Some literatures even treat stability as a multifaceted notion and resilience is one component of stability [797]. Since resilience itself is a multidimensional concept, integrating persistence, resistance and the ability to recover/adapt, we trace back to the original definitions of these two concepts [4] and point out that resilience and stability describe distinct properties of systems. The concept “resilience” concentrate on the boundaries to the domain of attraction while “stability” focus on equilibrium states. On one hand, a system can be Figure 81: Stability in network of microbiota. (A) Ecological very resilient and still fluctuate greatly, i.e. have low network theory captures networks of microbial species that interact with themselves (−s) and other genotypes (ai j). (B) Coupled stability. For instance, pest systems are highly variable ordinary differential equations capture all possible combinations of in space and time; as open systems, they are much af- connectivity, and interaction types. Three sample networks are fected by dispersal and therefore have a high resilience shown. (C) Communities that return to their previous densities after [798]. On the other hand, a stable system may have low perturbation are classified as stable, those that return to their equilibrium faster are categorized as more stable, and those that resilience. For example, the commercial fishery systems continue to diverge from the equilibrium are considered unstable of the Great Lakes are quite sensitive to disruption by [801]. (D) Linear stability analysis uses the eigenvalues’ real (Re) man, for they represent climatically buffered, fairly ho- and imaginary (Im) parts. Source: mogeneous and self-contained systems with relatively Figure from [802]. low variability and hence high stability and low resilience [4]. As shown in Fig. 81, there are different ways of sta- 6.3. Robustness in networks bility analysis. On one side, nonlinear dynamical sys- The term ‘robustness’ is derived from the Latin term tems are usually modelled as coupled ordinary differen- Quercus Robur that means oak, which was the sym- tial equations, and the nonlinear stability analysis can be bol of strength and longevity in the ancient world. In a realised by observing how the states of variables evolve system, robustness characterises its ability to withstand with time after perturbations. If the system could go failures and perturbations without loss of functions. For back to the original state before perturbations, then this instance, biologists define robustness as the ability of system is stable, and the faster the recovery is, the more living systems to maintain specific functionalities de- stable the system is. On the other side, system’s stability spite unpredictable perturbations [803]. Particularly, in can be analysed in a linear way, that is, by studying the network science, robustness is a concept related to the 85 network’s static topology, measuring its ability to maintain its connectivity when a fraction of nodes (links) is damaged [40, 804]. Mathematically, the robustness in a network is modeled as a reverse percolation process [805, 806, 807]. Percolation theory models the process of randomly placing pebbles to a square lattice with probability p, and predicts the critical value pc where a large cluster (called ‘percolating cluster’) emerges. At the critical point pc, a phase transition appears: many small clusters coalesce, forming a percolating cluster that perco- lates the whole lattice. For analysing the robustness of a network, we randomly remove a fraction f of nodes from the network and observe how the largest connected component size changes. When f is small, the node removal brings little damage to the network and a largest connected component continuously exists in the network. Once f reaches a critical point fc, the largest connected component van- ishes, as shown in Fig. 82. The robustness of a network is usually either characterised by the value of the critical threshold fc analysed using percolation theory or defined as the integrated size of the largest connected cluster during the entire attack process [647, 769], which is R 1 P d f . Figure 82: Network robustness is characterised by the size of giant 0 ∞ connected component after random removal of f fraction of nodes. For a random network with arbitrary degree distribu- Source: Figure from [2]. tions P(k), the largest connected component [808] exists if hk2i κ = > 2, (6.7) hki why some mutations lead to diseases and others do not in biology and medicine, and enable us to make the in- which is called the Molloy-Reed criterion [809,1]. The frastructures we use in everyday life more efficient and random removal of an f fraction of nodes leads to more robust. the changes in the degree distribution and parameter κ. Once the parameter κ < 2, the largest connected com- In real-world, systems are not isolate, but interdepen- ponent disappears and the network is fragmented into dent or interact with one another. Take the interactions many disconnected components. Based on the Molloy- between systems into consideration, many studies have Reed criterion, the critical percolation threshold follows been carried out to study the robustness in interdependent networks [14, 810, 811, 812, 813, 814, 815, 816, 1 648, 619, 817, 818], interconnected networks [819, 820, fc = 1 − , (6.8) hk2i 649], multi-layered networks [821, 822, 823, 824, 825], hki − 1 multiplex networks [826], and a network of networks where hk2i and hki are uniquely determined by the de- [647, 827, 828, 829]. In these systems, networks inter- gree distribution P(k). For the an Erdos-R˝ enyi´ (ER) net- act with each other and exhibit structural and dynami- work with average degree being hki, the critical thresh- cal features that differ from those observed in isolated ER old is fc = 1 − 1/hki. For a scale-free network with networks. For example, Buldyrev et al. [14] devel- −γ SF degree distribution P(k) ∼ k , the threshold fc → 1 oped an analytical framework based on the generating as N → ∞ if 2 < γ < 3, which means that we have to function formalism [830, 831], describing the cascad- remove all the nodes in order to fragment such network ing failures in two interdependent networks, and find- [1]. For example, the physical structure of the Internet ing a first order discontinuous phase transition, which (γ ≈ 2.5) is impressively robust, with pc > 0.99. The is dramatically different from the second order contin- study of the robustness of complex systems can help us uous phase transition found in isolated networks. Par- understand the real-world, for example, help understand shani et al. [810] studied a model more close to real 86 systems, two partially interdependent networks, finding [783, 836, 837], the dynamics are mainly limited to that the percolation transition changes from a first or- oscillations which are more related to synchronization der to a second order at a certain critical coupling as [782]. Thus, resilience is a broader concept for ana- the coupling strength decreases. Gao et al. developed lyzing the dynamical changes in networks’ response to an analytical framework to study the percolation of a perturbations. tree-like network formed by n interdependent networks In summary, robustness of a network is its abil- (tree-like NON) [646, 647], discovering that while for ity to maintain its statical topological integrity under n = 1 the percolation transition is a second order, but for node/link failures. The topological integrity of a net- any n > 1 where cascading failures occur, it is a first or- work is usually characterized by its connectivities, such der (abrupt) transition. Liu et al. [817] find hybrid phase as the diameter [838] or the size of the largest connected transition in the interdependent directed networks. component. The robustness of networks is also related to the failure mechanism. The studies reviewed above mostly fo- 7. Conclusions and future perspectives cus on random failure, while in real scenarios, initial failures are mostly not random but due to targeted attack 7.1. Conclusions on important hubs (nodes with high degree) or occur to Resilience has long been recognized as a defining low degree nodes since important hubs are purposely property of many dynamical complex systems [4], and protected [832]. Single real networks, like Internet, are network resilience has already become a new emerging vulnerable to targteted attacks [607, 833]. The simulta- sub-field in network science [12]. In this report, we have neous removal of several hubs will break any network. reviewed several primary theoretical tools and empiri- For coupled networks, Huang et al. [832] proposed a cal studies to analyze the resilience of both low- and mathematical framework for understanding the robust- high-dimensional systems. To quantify the resilience ness of fully interdependent networks under targeted at- of a complex system, we first need to uncover its dy- tacks, which was later extended to targeted attacks on namics, which are usually modeled as ordinary differen- partially interdependent networks by Dong et al. [834]. tial equations. For low-dimensional networks, we could Huang et al. [832] and Dong et al. [834] developed detect their attractors by calculating the fixed points of a general technique that uses the random attack solu- the equations [73, 91], and use the quasi-potential land- tion to map the solution onto the targeted attack prob- scape to show the trajectory of transitions between dif- lem in interdependent networks. Furthermore, Dong et ferent attractors [395, 321]. For high-dimensional net- al. [828] extended the study of targeted attacks on high works, in addition to the traditional linearization method degree nodes in a pair of interdependent networks to the [36, 33], dimension-reduction method based on mean- study of network of networks, and find that the robust- field theory has been verified as a powerful tool to pre- ness of networks of networks can be improved by pro- dict the tipping points [12, 54]; or we could use the tecting important hubs. changes in network structure as early warning signals In most studies of network robustness, networks are for the regime shifts [458, 472]. treated as static and unweighted [1, 835, 14]. How- We have incorporated the discussed above primary ever, such assumption is not applicable to real networks, theoretical tools into the discussion of their applications which are usually dynamical and weighted. For ex- and extensively reviewed state-of-the-art progress while ample, a traffic network is always topologically con- analyzing the resilience of ecological, biological, social, nected but maybe dynamically failed, since the flux and infrastructure systems. A series of topics, such as could be zero. In addition, the link weight is also the alternative stable states (also called multiple stable very important to network’s function. Take the inter- states), regime shifts, feedback loops, and early warn- actions between clownfish and sea anemone for exam- ing signals, have been illustrated in both mathematical ple, even if there are clownfishes and sea anemones models and empirical studies. We also emphasized the in the same pool and there are possible interactions impact of the network on resilience. We pointed out between them, most clownfishes could not find sea that network robustness and stability are distinct con- anemones when the water is seriously polluted, indi- cepts from network resilience, with network robustness cating a very low weight of their interactions. Loosing focusing on network topology, stability being a property the protection from sea anemones, clownfishes may be of system with a degree of fluctuation around a specific eaten quickly, and sea anemones could also die without fixed point, while resilience being a broader concept the food brought by clownfishes. Although there are with withstand, recover and adaptation being its three some studies on “dynamical robustness” of networks essential elements. 87 7.2. Future perspectives driving force for dynamic changes. Hence, a more general model represented by partial differential equations Given the rapid advances in the resilience of complex may be needed. Last, the time scale for each component networks, we have discussed the results from both the- may vary in a vast range from seconds to months, which oretical and empirical studies that will likely stay with brings significant challenges to dynamic modeling. A us for many years to come. As for any emerging field, feasible way for considering all these complexities is to there is a large body of relevant questions and upcoming combine enough prior knowledge about the real system developments that may further increase the relevance and the data-driven dynamic model discovering meth- of the discussed frameworks. Next, we highlight sev- ods. eral research topics being addressed to achieve a comprehensive understanding of the resilience of networked systems. 7.2.2. Empirical studies of resilience Despite the extensive studies on network resilience 7.2.1. Uncovering network dynamics form data on sys- and related topics, such as alternative stable states, tem evolution regime shifts, and feedback loops, there are far more theoretical studies than empirical studies. For exam- As we discussed in Chapter6, network dynamics is a ple, comparing to the plenty of mathematical models prerequisite for resilience analysis. Due to the complex- showing the alternative stable states [28], but there are ity and often unknown mechanisms operating within a few empirical studies, especially those conducted in complex networks, it is difficult to determine the equa- the wild world instead of experiments in laboratories. tion forms, and a set of parameters and mechanistic The reason is that real networks are usually too com- models are rare for complex systems. Thanks to the plex and interconnected for real world experimentation. accumulation of massive data, capturing the detailed There are always gaps between theoretical models and node-level dynamics of biological, social and techno- real systems, such as the universal dimension-deduction logical systems has become possible, which could help method for predicting the resilience of complex net- us peek into the inner mechanisms of complex systems. works is based on the assumption that there are no neg- It has also become trendy to develop tools for discover- ative interactions in the networks [12]. ing the model of network dynamic directly from data The remaining challenges in this field include bridg- [52, 839], or just network topology [840, 841] even ing the gaps between the theoretical model and real net- without a mechanistic model. Previous studies have works and applying the resilience theory to various real- shown that a complex system’s response to perturba- world systems, such as traffic networks, power grid, hu- tions is driven by a small number of universal mecha- man mobility network [843, 844], ecological and bio- nisms [242]. Thus, the authors also proposed a method logical systems. For these real-world systems, it is cru- to infer the microscopic dynamics of a complex system cial to develop strategies to improve their resilience and from observations of its response to external perturba- prevent collapse. Especially, nowadays there are mas- tions [52], allowing us to construct an effective dynamic sive data on these systems. Predicting the tipping point model. By separating the contribution of network topol- in these systems directly on data rather than models re- ogy from the dynamics, Barzel et al. [840, 841] devel- mains an open problem. Besides, previous studies treat oped general tools to translate a network’s topology into the structure and dynamics independently to see how predictions of its observed propagation patterns. the structure perturbations affect the system’s resilience. Despite these excellent efforts, tools for uncovering However, there is a “adaptive cycle” in real-world sys- real dynamics for a specific real system are still lack- tems [23], with which the system will adjust its dynam- ing. Firstly, these studies assume that the nodes in the ics to compensate for structure perturbations. It remains same network follow the same dynamical pattern, while to be a challenge to reveal the interplay between struc- in a real network, nodes may have very different dy- ture and dynamics. namical behavior. For example, in a protein interaction network, two proteins may be connected because of simple physical binding/separation or due to the fact 7.2.3. Resilience of networks of networks they participate in the same metabolic reaction to pro- It is increasingly clear that networks in the real world duce other compounds [842]. Secondly, the dynamic are not isolated but interdependent with one another. models discovered in these studies are all ordinary dif- For example, for diverse critical infrastructures, the ferential equations with time being the argument. While electric power network provides power for pumping and other variables, such as spatial distance, could be a vital for control systems of the water network. In turn, the 88 water network provides water for cooling and emis- hensive analytical framework is needed to predict the sions reduction of the power network, the fuel network true dynamics behavior and resilience of the original provides fuel for generators for electric power network system using the incomplete network. This framework and the electric power network provides power to pump must contain an unbiased graph sampling method that oil for fuel network [769]. A set of networks may preserves the dynamics of the sampled nodes experi- couple together, forming multilayer networks [845], or enced by nodes in the original network. It could facil- networks of networks [846]. As discussed in Chap- itate other related studies on large-scale dynamical net- ter6, studies focusing mainly on the structural integrity works, such as the controllability analysis [48, 855] and and robustness have been carried out already [14, 825]. studies of coevolution spreading dynamics [856]. Based In contrast, resilience is a property related to network on such a framework, future studies on how to predict dynamics, and these networks may be characterized the dynamics and resilience of incomplete networks of by different time scales and structural patterns, which networks could be another essential branch of network makes the analysis of resilience of networks of networks resilience. extremely difficult. Early attempts have focused on specific dimension-reduction methods for ecological mutualistic networks [54] or purely numerical studies [845]. 7.2.5. Controlling the resilience of networks Despite these efforts, it remains a challenge to develop A reflection of our ultimate understanding of a com- a general framework for analyzing the resilience of net- plex system is our ability to control its behavior [48]. It works of networks, which could trigger a series of future requires an accurate model for network topology and research. dynamics and powerful tools for analyzing the network’s resilience. Early related studies on controlling 7.2.4. Resilience of networks with incomplete informa- complex networks have been focused on the controlla- tion bility of linear and nonlinear systems [756, 857] or con- Most real networks are incomplete due to the com- trolling collective behavior in complex networks [858]. plexity and uncovered knowledge about the original The latter studies ignored the strength of interactions be- systems, such as hidden relationships in social networks tween dynamical components, making them unsuitable [847], unmeasured connections in biological networks for controlling network resilience here. As discussed in [848], and adaptively changing interactions between Chapter.2, Jiang et al. [239] investigate how to man- species in ecological networks [849]. It is essential to age or control tipping points in real-world complex and develop tools to infer the original network structure and nonlinear dynamical networks in ecology, by altering dynamics from incomplete information. Even a branch the way that species extinction occurs: from massive in network science, called link prediction, which aims extinction of all species to the gradual extinction of in- to uncover the hidden relationships [848, 850], could dividual species, so that the occurrence of global ex- move a step towards this goal. It still would be difficult tinction is substantially delayed. It is realized by merely to achieve this goal fully. The more important challenge fixing the abundances of some species, and the goal is is that real complete networks are usually too large to to delay the tipping point. handle. For example, the social network, Facebook, has Controlling network resilience is extremely difficult attracted more than 2.5 billion users 2019. Hence, an but deserves our efforts, because it is critical for in- inverse problem has attracted much attention, that is, frastructural networks, with real and immediate need to given a huge real network, how can we derive a repre- control the dynamics of information flow, and for so- sentative sample [851]? Graph sampling is a technique cial and organizational networks, with the ongoing need to create a small but representative sample (an incom- to understand the control principles of adversary net- plete network) by picking a subset of nodes and links works. We could move towards such a goal by devel- from the original network [852]. Serval sampling meth- oping theory for quantifying the safe operating space ods, such as random walk [853], breadth-first sampling [859], and proposing strategies to enhance network re- [854], and scaling-down sampling [851], has been pro- silience and recoverability in networks and multilayer posed for graph sampling and has been proved to be able networks. The strategies could include node state in- to keep the network scale small while capturing specific tervention, addition/diminution of links, rewiring, and properties of the original network. weight changes. The resilience control and optimiza- Unfortunately, all the current research about graph tion represent a new direction in network science, offer- sampling mainly consider the topological properties, ig- ing new avenues to control the dynamical behavior of noring the nonlinear dynamics of networks. A compre- many real-world systems. 89 References logical resilience. 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