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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 transi- tions 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 sys- tems 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 im- pacts through situation assessment, rapid response, and active recovery strategies [20]. Generally, these defi- nitions vary between three extremes: system resilience (or called ecological resilience) that focus on the magni- tude of the change or perturbation that a system can en- dure without shifting into another stable state [4], engi- neering 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].
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