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Robustness of network of networks under targeted attack

Gaogao Dong1,2,*, Jianxi Gao2,3, Ruijin Du1,2,4, Lixin Tian1, H. Eugene Stanley2, and Shlomo Havlin5

  • 1Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China
  • 2Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
  • 3Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
  • 4College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
  • 5Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel

  • *dfocus.gao@gmail.com
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Phys. Rev. E 87, 052804 – Published 16 May, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.052804

Abstract

The robustness of a network of networks (NON) under random attack has been studied recently [Gao et al., Phys. Rev. Lett. 107, 195701 (2011)]. Understanding how robust a NON is to targeted attacks is a major challenge when designing resilient infrastructures. We address here the question how the robustness of a NON is affected by targeted attack on high- or low-degree nodes. We introduce a targeted attack probability function that is dependent upon node degree and study the robustness of two types of NON under targeted attack: (i) a tree of n fully interdependent Erdős-Rényi or scale-free networks and (ii) a starlike network of n partially interdependent Erdős-Rényi networks. For any tree of n fully interdependent Erdős-Rényi networks and scale-free networks under targeted attack, we find that the network becomes significantly more vulnerable when nodes of higher degree have higher probability to fail. When the probability that a node will fail is proportional to its degree, for a NON composed of Erdős-Rényi networks we find analytical solutions for the mutual giant component P∞ as a function of p, where 1−p is the initial fraction of failed nodes in each network. We also find analytical solutions for the critical fraction pc, which causes the fragmentation of the n interdependent networks, and for the minimum average degree k¯min below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdős-Rényi networks under targeted attack, we find the critical coupling strength qc for different n. When q>qc, the attacked system undergoes an abrupt first order type transition. When q≤qc, the system displays a smooth second order percolation transition. We also evaluate how the central network becomes more vulnerable as the number of networks with the same coupling strength q increases. The limit of q=0 represents no dependency, and the results are consistent with the classical percolation theory of a single network under targeted attack.

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Phys. Rev. E 87, 052804– Published 16 May, 2013
  • Received 12 February 2013
Article part of CHORUS

©2013 American Physical Society

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