Percolation of interdependent networks with intersimilarity

摘要

Real data show that interdependent networks usually involve intersimilarity. Intersimilarity means that a pairnof interdependent nodes have neighbors in both networks that are also interdependent [Parshani et al. Europhys.nLett. 92, 68002 (2010)]. For example, the coupled worldwide port network and the global airport network arenintersimilar since many pairs of linked nodes (neighboring cities), by direct flights and direct shipping lines, existnin both networks. Nodes in both networks in the same city are regarded as interdependent. If two neighboringnnodes in one network depend on neighboring nodes in the other network, we call these links common links. Thenfraction of common links in the system is a measure of intersimilarity. Previous simulation results of Parshani et al.nsuggest that intersimilarity has considerable effects on reducing the cascading failures; however, a theoreticalnunderstanding of this effect on the cascading process is currently missing. Here we map the cascading processnwith intersimilarity to a percolation of networks composed of components of common links and noncommonnlinks. This transforms the percolation of intersimilar system to a regular percolation on a series of subnetworks,nwhich can be solved analytically. We apply our analysis to the case where the network of common links is annErd˝os-R´enyi (ER) network with the average degree K, and the two networks of noncommon links are also ERnnetworks.We show for a fully coupled pair of ER networks, that for any K u0002 0, although the cascade is reducednwith increasing K, the phase transition is still discontinuous. Our analysis can be generalized to any kind ofninterdependent random network systems.