TY - JOUR

T1 - Robustness of network of networks under targeted attack

AU - Dong, Gaogao

AU - Gao, Jianxi

AU - Du, Ruijin

AU - Tian, Lixin

AU - Stanley, H. Eugene

AU - Havlin, Shlomo

PY - 2013/5/16

Y1 - 2013/5/16

N2 - The robustness of a network of networks (NON) under random attack has been studied recently. 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 Erdos-Rényi or scale-free networks and (ii) a starlike network of n partially interdependent Erdos-Rényi networks. For any tree of n fully interdependent Erdos-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 Erdos-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 k̄min below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdos-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.

AB - The robustness of a network of networks (NON) under random attack has been studied recently. 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 Erdos-Rényi or scale-free networks and (ii) a starlike network of n partially interdependent Erdos-Rényi networks. For any tree of n fully interdependent Erdos-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 Erdos-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 k̄min below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdos-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.

UR - http://www.scopus.com/inward/record.url?scp=84878373224&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.87.052804

DO - 10.1103/PhysRevE.87.052804

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C2 - 23767581

AN - SCOPUS:84878373224

SN - 1539-3755

VL - 87

JO - Physical Review E

JF - Physical Review E

IS - 5

M1 - 052804

ER -