Robustness of network of networks under targeted attack

摘要

The robustness of a network of networks (NON) under random attack has been studied recently [Gao et al.,nPhys. Rev. Lett. 107, 195701 (2011)]. Understanding how robust a NON is to targeted attacks is a major challengenwhen designing resilient infrastructures. We address here the question how the robustness of a NON is affectednby targeted attack on high- or low-degree nodes. We introduce a targeted attack probability function that isndependent upon node degree and study the robustness of two types of NON under targeted attack: (i) a tree of nnfully interdependent Erd˝os-R´enyi or scale-free networks and (ii) a starlike network of n partially interdependentnErd˝os-R´enyi networks. For any tree of n fully interdependent Erd˝os-R´enyi networks and scale-free networksnunder targeted attack, we find that the network becomes significantly more vulnerable when nodes of higherndegree have higher probability to fail. When the probability that a node will fail is proportional to its degree,nfor a NON composed of Erd˝os-R´enyi networks we find analytical solutions for the mutual giant component P∞nas a function of p, where 1 − p is the initial fraction of failed nodes in each network. We also find analyticalnsolutions for the critical fraction pc, which causes the fragmentation of the n interdependent networks, and for thenminimum average degree ¯kmin below which the NON will collapse even if only a single node fails. For a starlikenNON of n partially interdependent Erd˝os-R´enyi networks under targeted attack, we find the critical couplingnstrength qc for different n. When q > qc , the attacked system undergoes an abrupt first order type transition.nWhen q u0002 qc, the system displays a smooth second order percolation transition.We also evaluate how the centralnnetwork becomes more vulnerable as the number of networks with the same coupling strength q increases. Thenlimit of q = 0 represents no dependency, and the results are consistent with the classical percolation theory of ansingle network under targeted attack.