title: Robustness of network of networks under targeted attack.
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 p(c), which causes the fragmentation of the n interdependent networks, and for the minimum average degree k[over ¯](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 q(c) for different n. When q>q(c), the attacked system undergoes an abrupt first order type transition. When q≤q(c), 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.
authors.affiliation: Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China. dfocus.gao@gmail.com
authors.forename: Gaogao
authors.initials: G
authors.lastname: Dong
authors.forename: Jianxi
authors.initials: J
authors.lastname: Gao
authors.forename: Ruijin
authors.initials: R
authors.lastname: Du
authors.forename: Lixin
authors.initials: L
authors.lastname: Tian
authors.forename: H Eugene
authors.initials: HE
authors.lastname: Stanley
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin
mesh_terms: D000465:Algorithms; D003195:Computer Communication Networks; D016494:Computer Security; D003198:Computer Simulation; D008962:Models, Theoretical