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 $P_{\infty }$ 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 ${\overline{k}}_{\min }$ 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 $q\le q_c$, 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.

• Received 12 February 2013

DOI:https://doi.org/10.1103/PhysRevE.87.052804