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 Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi or scale-free networks and (ii) a starlike network of $n$ partially interdependent Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi networks. For any tree of $n$ fully interdependent Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi 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 Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi 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 ${\overline{k}}_{\mathrm{min}}$ below which the NON will collapse even if only a single node fails. For a starlike NON of $n$ partially interdependent Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi 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.