Percolation of a general network of networks
Percolation of a general network of networks
复制标题
DOI：
10.1103/physreve.88.062816
复制
发表时间：
2013-12-20
影响因子：
2.4
通讯作者：
Havlin, Shlomo
中科院分区：
文献类型：
--
作者：
Gao, Jianxi;Buldyrev, Sergey V.;Havlin, Shlomo
逾渗理论是研究系统脆弱性的一种方法。我们开发了一个分析框架，并分析了由相互依赖的网络（NetONet）组成的网络的渗流特性。典型地，单个网络的逾渗表明，由于故障而导致的网络中的损坏是故障大小的连续函数，即，故障节点的比例。与此形成鲜明对比的是，在NetONet中，由于级联故障，渗流过渡可能是不连续的，甚至单个节点故障可能导致系统的突然崩溃。我们展示了由n个经典的Erdos-Renyi（ER）网络组成的NetONet的一般框架，其中每个网络依赖于相同数量的其他网络，即，对于由相互依赖的ER网络形成的随机规则网络（RR）。不同网络的节点之间的依赖性被视为一一对应，即，一个网络中的节点可以仅依赖于另一个网络中的一个节点（无反馈条件）。与最大连接簇（互分量）的大小取决于n的树状NetONet不同，RR NetONet中的循环导致最大连接簇仅取决于m和每个网络的拓扑结构，而不取决于n。我们还分析了极脆弱的耦合反馈条件，其中不同网络节点之间的耦合不是一一对应的。在由ER网络构成的NetONet中，渗流只表现出二级相变和坍缩两个阶段，而在无反馈条件下，没有发现一级渗流相变。在由RR网络组成的NetONet中，当耦合强度q（相互依赖链路的分数）较大时，存在一阶相变，当q较小时，存在二阶相变。我们对耦合网络弹性的洞察可能有助于设计强大的相互依赖系统。
Percolation theory is an approach to study the vulnerability of a system. We develop an analytical framework and analyze the percolation properties of a network composed of interdependent networks (NetONet). Typically, percolation of a single network shows that the damage in the network due to a failure is a continuous function of the size of the failure, i.e., the fraction of failed nodes. In sharp contrast, in NetONet, due to the cascading failures, the percolation transition may be discontinuous and even a single node failure may lead to an abrupt collapse of the system. We demonstrate our general framework for a NetONet composed of n classic Erdos-Renyi (ER) networks, where each network depends on the same number m of other networks, i.e., for a random regular network (RR) formed of interdependent ER networks. The dependency between nodes of different networks is taken as one-to-one correspondence, i.e., a node in one network can depend only on one node in the other network (no-feedback condition). In contrast to a treelike NetONet in which the size of the largest connected cluster (mutual component) depends on n, the loops in the RR NetONet cause the largest connected cluster to depend only on m and the topology of each network but not on n. We also analyzed the extremely vulnerable feedback condition of coupling, where the coupling between nodes of different networks is not one-to-one correspondence. In the case of NetONet formed of ER networks, percolation only exhibits two phases, a second order phase transition and collapse, and no first order percolation transition regime is found in the case of the no-feedback condition. In the case of NetONet composed of RR networks, there exists a first order phase transition when the coupling strength q (fraction of interdependency links) is large and a second order phase transition when q is small. Our insight on the resilience of coupled networks might help in designing robust interdependent systems.