Robustness of network of networks under targeted attack

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 p_c, 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 Erdos-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.