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System Crash As Dynamics of Complex Networks

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System Crash As Dynamics of Complex Networks System crash as dynamics of complex networks Yi Yua,1, Gaoxi Xiaoa,b,1,2, Jie Zhouc, Yubo Wanga, Zhen Wangd,e,1,2, Jürgen Kurthsf,g,h, and Hans Joachim Schellnhuberf,2 aSchool of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798; bComplexity Institute, Nanyang Technological University, Singapore 639798; cDepartment of Physics, East China Normal University, Shanghai 200241, China; dQingdao University, Qingdao, Shandong 266071, China; eInterdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan; fPotsdam Institute for Climate Impact Research, 14473 Potsdam, Germany; gDepartment of Physics, Humboldt University, 12489 Berlin, Germany; and hInstitute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom Contributed by Hans Joachim Schellnhuber, August 11, 2016 (sent for review June 8, 2016; reviewed by Ying-Cheng Lai and Matjaz Perc) Complex systems, from animal herds to human nations, sometimes to explain a sudden crash when (and only when) most indi- crash drastically. Although the growth and evolution of systems have viduals are close to the borderline between alternatives; in- been extensively studied, our understanding of how systems crash is formational cascade theories (18, 19) point out that, if still limited. It remains rather puzzling why some systems, appearing individuals follow the behavior of the preceding individuals to be doomed to fail, manage to survive for a long time whereas without regarding their own information, a big shift in the some other systems, which seem to be too big or too strong to fail, system may be caused by a small shock; and studies on lock-in crash rapidly. In this contribution, we propose a network-based effects and switching cost (20, 21), which evaluate effects of system dynamics model, where individual actions based on the local the cost that individuals have to pay to change a choice, may information accessible in their respective system structures may lead help explain the existence of a pseudo-steady state. Such “ ” to the peculiar dynamics of system crash mentioned above. Exten- studies, however, have their respective limitations: Conformity sive simulations are carried out on synthetic and real-life networks, theories in themselves cannot explain how individuals get close which further reveal the interesting system evolution leading to the to the borderline; informational cascade theories may not work final crash. Applications and possible extensions of the proposed well for system crashes where individuals do not easily disre- model are discussed. gard their own information; and the critical question of how individuals overcome the lock-in effects in the system crash complex systems | system crash | pseudo-steady state | cascade behavior remains largely open. Although most of the above studies have largely ignored ef- ystems emerge and grow up, and, naturally, systems die. Al- fects of the underlying structures of connections (22), an im- Sthough extensive studies have been carried out on the growth portant trend in research on complex systems is to study them in and evolution of various complex systems, in many cases the content of various complex networks. The most notable adopting complex network-based approaches (e.g., refs. 1 and 2), studies that combine system crash dynamics and network analysis hardly any similar studies exist on system crash. As a special type are probably those on k-core cascade theory (11, 23). Specifi- of system evolution, however, system crash defies straightforward cally, the k-core of a network is defined as the maximum subset description by any of the existing models of system evolution, to the of the network where each node is connected to at least k other best of our knowledge. A few important observations of system “ ” nodes in the same subset. Assume that a network node with crash remain puzzling and require careful study: (i) Some inferior fewer than k connections may have a chance to leave the current or “outdated” systems, although appearing to be doomed to fail, “ ” system [e.g., an OSN user may need to have a large enough may survive for a very long time. Examples include living fossils number of connections to justify the effort of staying on (11)]. such as coelacanths, which have survived for more than 80 million years (3), some depleted-but-not-eliminated species under compe- Significance tition or faced by seemingly unstoppable invasion (4), and social systems such as the final Qing dynasty in China, which went through 70 y of disastrous defeats and rebellions with surprising robustness System crash, as an essential part of system evolution, sometimes until its sudden termination in 1911 (5). We term such systems happens in peculiar manners: Weakened systems may survive for as being in a “pseudo-steady state” before their final crashes. a surprisingly long time before suddenly meeting their final ends, (ii) Systems that appear to be too big or too strong to fail whereas seemingly unbeatable giants may drastically crash to sometimes crash rapidly. Well-known examples include the virtual nonexistence. We propose a model that describes system crashes of various ecological and biological systems, which, in crash as a consequence of some relatively simple local in- some cases, occur much faster than expected (6, 7) or even for formation-based individual behaviors: Individuals leave networks no obvious reason (8, 9). Similar phenomena exist in human according to some most straightforward assessment of current society, e.g., the sudden crash of the Soviet Union (10) and the and future benefits/risks. Of note, such a simple rule may enable crash of the once biggest online social network (OSN), a single push/mistake to cause multistage-style system crash. Friendster, in less than 1 y (11). To help understand the Our study helps to make sense of the process where complex crashes of ecological (6, 7, 12), biological (8, 9), neurological systems go into unstoppable cascading declines and provides (13), physical and cyberphysical (14), economic (15, 16), social a viewpoint of predicting the fate of some social/natural (5, 10, 11), and many other complex systems, new modeling ap- systems. proaches to describe the process and dynamics of a system crash Author contributions: Y.Y., G.X., J.Z., Y.W., Z.W., J.K., and H.J.S. designed research; Y.Y., are needed. G.X., J.Z., Y.W., and Z.W. performed research; and Y.Y., G.X., Z.W., J.K., and H.J.S. wrote Some relevant studies, although not focusing directly on sys- the paper. tem crash, may help in understanding a few aspects of system Reviewers: Y.-C.L., Arizona State University; and M.P., University of Maribor. crash or system crash under some special cases, and are hence The authors declare no conflict of interest. worth mention. The largest class of relevant studies is probably 1Y.Y., G.X., and Z.W. contributed equally to this work. those on herd behavior, which refers to system behavior where 2To whom correspondence may be addressed. Email: [email protected], zhenwang0@ individuals in a group act collectively without centralized control gmail.com, or [email protected]. – (17 22). For example, conformity theories (17), which focus on This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. exploring the mechanism leading to uniform behavior, may help 1073/pnas.1612094113/-/DCSupplemental. 11726–11731 | PNAS | October 18, 2016 | vol. 113 | no. 42 www.pnas.org/cgi/doi/10.1073/pnas.1612094113 Downloaded by guest on September 29, 2021 dynamics. For example, although it is difficult to provide a rigid proof, in our extensive numerical simulations on various synthetic and real-life networks, neither the k-cascade model nor the q-cascade model has ever allowed the emergence of the pseudo- steady state. The proposed model is tested on various synthetic and real-life networks. Considering the increasing importance of OSNs and the relative easiness in getting the data of OSN structures, the testing on real-life networks will mainly focus on OSNs, although the proposed model certainly applies to many different kinds of complex systems. For the well-known case of Friendster’s quick crash, where there exist relatively abundant data on the whole Fig. 1. Schematic illustration of KQ cascade with ks = 3 and q = 0.5. In a classic k-core cascade, the four nodes within the yellow square forming up a process (11), we perform individual-based simulations to mimic three-core would stay on while other nodes would leave. In the KQ cascade, the actions of each individual. A good match is achieved with however, because node 1 has lost more than 50% of its original neighbors, it some interesting insights. will leave in the next time step, which leaves each of the three nodes within the blue triangle with fewer than three connections, leading to the final Results crash of the network. The KQ-Cascade Model. We start with an original network GðV, EÞ, where V is the set of vertices and E is the set of edges. To keep the KQ-cascade model simple, we introduce only two key pa- Measuring the k-core of a network therefore estimates how many rameters: the critical degree k and the loss-tolerance coefficient nodes may stay on when the value of k increases due to various s q. In the ith time step, any node with a degree k < ks or having reasons. This theory, however, can only explain a sudden system lost more than q proportion of its original connections may leave crash when there exists a big jump in the value k due to the network with a probability fiðkÞ. some special reasons. Other interesting developments include An example of the KQ-cascade model is illustrated in Fig.
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