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

最近研究了网络之网络（NON）在随机攻击下的鲁棒性[高等人，《物理评论快报》107卷，195701（2011年）]。在设计具有弹性的基础设施时，理解网络之网络对针对性攻击的鲁棒性是一项重大挑战。我们在此探讨对高度节点或低度节点进行针对性攻击如何影响网络之网络的鲁棒性这一问题。我们引入一个取决于节点度的针对性攻击概率函数，并研究两种类型的网络之网络在针对性攻击下的鲁棒性：（i）由n个完全相互依赖的厄多斯 - 仁伊（Erdos - Renyi）网络或无标度网络构成的树状网络；（ii）由n个部分相互依赖的厄多斯 - 仁伊网络构成的星状网络。对于在针对性攻击下由n个完全相互依赖的厄多斯 - 仁伊网络和无标度网络构成的任何树状网络，我们发现当高度节点失效的概率更高时，网络变得明显更脆弱。当一个节点失效的概率与其度成正比时，对于由厄多斯 - 仁伊网络组成的网络之网络，我们找到了相互巨连通分支\(P_{∞}\)作为\(p\)的函数的解析解，其中\(1 - p\)是每个网络中初始失效节点的比例。我们还找到了临界比例\(p_{c}\)（它导致n个相互依赖的网络破碎）以及最小平均度\(\overline{k}_{min}\)（即使只有一个节点失效，网络之网络在其之下也会崩溃）的解析解。对于在针对性攻击下由n个部分相互依赖的厄多斯 - 仁伊网络构成的星状网络之网络，我们找到了不同\(n\)对应的临界耦合强度\(q_{c}\)。当\(q > q_{c}\)时，受攻击的系统会经历突然的一阶类型转变。当\(q\)（此处原文似乎不完整）