Gao Jianxi, Buldyrev S V, Havlin S, Stanley H E
Department of Automation, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, PR China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6 Pt 2):066134. doi: 10.1103/PhysRevE.85.066134. Epub 2012 Jun 29.
Many real-world networks interact with and depend upon other networks. We develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks, each composed of the same number of nodes N. The dependency links connecting nodes from different networks establish a unique one-to-one correspondence between the nodes of one network and the nodes of the other network. We study the dynamics of the cascades of failures in such a network of networks (NON) caused by a random initial attack on one of the networks, after which a fraction p of its nodes survives. We find for the fully interdependent loopless NON that the final state of the NON does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the NON, which generalizes both the giant component of regular percolation of a single network (n=1) and the recently studied case of the mutual giant component of two interdependent networks (n=2). We also find that the mutual giant component does not depend on the topology of the NON and express it in terms of generating functions of the degree distributions of the network. Our results show that, for any n≥2 there exists a critical p=p(c)>0 below which the mutual giant component abruptly collapses from a finite nonzero value for p≥p(c) to zero for p<p(c), as in a first-order phase transition. This behavior holds even for scale-free networks where p(c)=0 for n=1. We show that, if at least one of the networks in the NON has isolated or singly connected nodes, the NON completely disintegrates for sufficiently large n even if p=1. In contrast, in the absence of such nodes, the NON survives for any n for sufficiently large p. We illustrate this behavior by comparing two exactly solvable examples of NONs composed of Erdős-Rényi (ER) and random regular (RR) networks. We find that the robustness of n coupled RR networks of degree k is dramatically higher compared to the n-coupled ER networks of the same average degree k[over ¯]=k. While for ER NONs there exists a critical minimum average degree k[over ¯]=kover ¯∼lnn below which the system collapses, for RR NONs k(min)=2 for any n (i.e., for any k>2, a RR NON is stable for any n with p(c)<1). This results arises from the critical role played by singly connected nodes which exist in an ER NON and enhance the cascading failures, but do not exist in a RR NON.
许多现实世界中的网络与其他网络相互作用并相互依赖。我们开发了一个分析框架，用于研究由(n)个完全相互依赖的随机连接网络组成的网络，每个网络都由相同数量的节点(N)组成。连接不同网络节点的依赖链路在一个网络的节点与另一个网络的节点之间建立了唯一的一一对应关系。我们研究了由对其中一个网络的随机初始攻击引发的此类网络网络（NON）中的故障级联动态，在此之后，其节点的一部分(p)得以幸存。我们发现，对于完全相互依赖的无环NON，NON的最终状态不取决于级联的动态，而是由NON的唯一确定的相互巨型组件决定，该组件既推广了单个网络（(n = 1)）的常规渗流巨型组件，也推广了最近研究的两个相互依赖网络（(n = 2)）的相互巨型组件。我们还发现，相互巨型组件不依赖于NON的拓扑结构，并根据网络度分布的生成函数来表示它。我们的结果表明，对于任何(n\geq2)，存在一个临界值(p = p(c)>0)，低于该值时，相互巨型组件会从(p\geq p(c))时的有限非零值突然坍塌为(p < p(c))时的零值，就像在一阶相变中一样。即使对于(n = 1)时(p(c)=0)的无标度网络，这种行为也成立。我们表明，如果NON中的至少一个网络具有孤立或单连通节点，那么对于足够大的(n)，即使(p = 1)，NON也会完全解体。相反，在没有此类节点的情况下，对于足够大的(p)，NON对于任何(n)都能幸存。我们通过比较由厄多斯 - 雷尼（ER）网络和随机正则（RR）网络组成的两个可精确求解的NON示例来说明这种行为。我们发现，度为(k)的(n)个耦合RR网络的鲁棒性比具有相同平均度(\overline{k}=k)的(n)个耦合ER网络要高得多。虽然对于ER NON，存在一个临界最小平均度(\overline{k}=\overline{k}(min)\sim\ln n)，低于该值时系统会崩溃，但对于RR NON，对于任何(n)（即对于任何(k>2)，RR NON对于任何(n)且(p(c)<1)都是稳定的），(k(min)=2)。这种结果源于单连通节点所起的关键作用，单连通节点存在于ER NON中并增强了级联故障，但不存在于RR NON中。
