Robustness of network of networks under targeted attack

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 Erdos-Renyi or scale-free networks and (ii) a starlike network of n partially interdependent Erdos-Renyi networks. For any tree of n fully interdependent Erdos-Renyi 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-Renyi networks we find analytical solutions for the mutual giant component P-infinity 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 bar (min) below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdos-Renyi 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. DOI: 10.1103/PhysRevE.87.052804