Cascading dynamics on random networks: Crossover in phase transition

Run-Ran Liu, Wen-Xu Wang, Ying-Cheng Lai, and Bing-Hong Wang

Phys. Rev. E 85, 026110 – Published 16 February 2012

Abstract

In a complex network, random initial attacks or failures can trigger subsequent failures in a cascading manner, which is effectively a phase transition. Recent works have demonstrated that in networks with interdependent links so that the failure of one node causes the immediate failures of all nodes connected to it by such links, both first- and second-order phase transitions can arise. Moreover, there is a crossover between the two types of transitions at a critical system-parameter value. We demonstrate that these phenomena can occur in the more general setting where no interdependent links are present. A heuristic theory is derived to estimate the crossover and phase-transition points, and a remarkable agreement with numerics is obtained.

Received 11 February 2011

DOI:https://doi.org/10.1103/PhysRevE.85.026110

Authors & Affiliations

Run-Ran Liu^1,2,3,*, Wen-Xu Wang^3,4,†, Ying-Cheng Lai^3,5,‡, and Bing-Hong Wang^2,§

¹Institute for Information Economy, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
²Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei, Anhui 230026, China
³School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
⁴Department of Systems Science, School of Management and Center for Complexity Research, Beijing Normal University, Beijing 100875, China
⁵Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

^*runranliu@gmail.com
^†wenxuw@gmail.com
^‡Ying-Cheng.Lai@asu.edu
^§bhwang@ustc.edu.cn

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Issue

Vol. 85, Iss. 2 — February 2012

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Images

Figure 1
Simulation results demonstrating first- and second-order phase transitions on ER networks. Shown in panel (a) are the fraction of nodes in the giant component after cascade, denoted by $S_{1}$ , as a function of $p$ for $β = 0.3$ , $0.49$ , $0.51$ , and $0.7$ , respectively, where the network size is $10^{5}$ . Shown in panel (b) are the log-log plot of $S_{1}$ vs the distance between $p$ and the second-order phase-transition point $p^{I I}$ for different network sizes when $β = 0.49$ . Shown in panel (c) are the curves of $S_{1}$ vs $p$ depending on the network size $N$ for $β = 0.51$ . All the results are obtained by averaging over 1000 independent network realizations, and the average degree is $〈 k 〉 = 4$ .Reuse & Permissions
Figure 2
(a) For $β = 0.6$ (first-order phase transition), normalized size $S$ of the final fraction of preserved nodes (dotted line), normalized size of the giant component $S_{1}$ (dashed line) in the steady state, and NOI (full line) as a function of $p$ . (b) For $β = 0.3$ (second-order phase transition), $S$ (dotted line), $S_{1}$ (dashed line), and the normalized size of the second largest component $S_{2}$ (full line) in the steady state versus $p$ . Panels (c) and (d) display the scaling behaviors of NOI and $S_{1}$ for different parameter values of $β$ at criticality, respectively. The results are obtained by averaging over 1000 independent network realizations, where the average degree of networks is 4 and the network size is $10^{5}$ for panels (a) and (b).Reuse & Permissions
Figure 3
Diagram of the phase-transition points, $p^{I}$ in the first-order region (black dot) and $p^{I I}$ in the second-order region (red square) as a function of nodal vulnerability $β$ for ER networks with different values of $〈 k 〉$ : 4, 6, 8, and 10 for panels (a)–(d), respectively. The striped region denotes the second-order regime predicted by our theory. The simulation results are obtained by averaging over 1000 independent network realizations and the error bars are comparable to the size of the symbols.Reuse & Permissions
Figure 4
For $〈 k 〉 = 4$ , illustrations of graphical solutions of Eq. (3). (a) First-order phase transition ( $β = 0.55$ ), where the black dot denotes the tangent point. (b) Second-order phase transition ( $β = 0.5$ ).Reuse & Permissions
Figure 5
Comparison between the numerically obtained first-order phase-transition points and theoretical predictions for ER networks of different average degree $〈 k 〉$ and nodal vulnerabilities $β$ . The symbols denote the numerical results and the lines are from theory. The simulation results are obtained by averaging over 1000 independent network realizations and the error bars are comparable to the size of the symbols, where the network size is $10^{5}$ .Reuse & Permissions

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