title: Percolation of interdependent networks with intersimilarity.
abstract: Real data show that interdependent networks usually involve intersimilarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent [Parshani et al. Europhys. Lett. 92, 68002 (2010)]. For example, the coupled worldwide port network and the global airport network are intersimilar since many pairs of linked nodes (neighboring cities), by direct flights and direct shipping lines, exist in both networks. Nodes in both networks in the same city are regarded as interdependent. If two neighboring nodes in one network depend on neighboring nodes in the other network, we call these links common links. The fraction of common links in the system is a measure of intersimilarity. Previous simulation results of Parshani et al. suggest that intersimilarity has considerable effects on reducing the cascading failures; however, a theoretical understanding of this effect on the cascading process is currently missing. Here we map the cascading process with intersimilarity to a percolation of networks composed of components of common links and noncommon links. This transforms the percolation of intersimilar system to a regular percolation on a series of subnetworks, which can be solved analytically. We apply our analysis to the case where the network of common links is an Erdős-Rényi (ER) network with the average degree K, and the two networks of noncommon links are also ER networks. We show for a fully coupled pair of ER networks, that for any K≥0, although the cascade is reduced with increasing K, the phase transition is still discontinuous. Our analysis can be generalized to any kind of interdependent random network systems.
authors: School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China and Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA.
authors: Yanqing
authors: Y
authors: Hu
authors: Department of Systems Science, Beijing Normal University, Beijing 100875, China and Physics Department, Bar-Ilan University, Ramat Gan 52900, Israel.
authors: Dong
authors: D
authors: Zhou
authors: Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA.
authors: Rui
authors: R
authors: Zhang
authors: Department of Systems Science, Beijing Normal University, Beijing 100875, China.
authors: Zhangang
authors: Z
authors: Han
authors: Institute of Geography, University of Lausanne, Lausanne 1015, Switzerland.
authors: Céline
authors: C
authors: Rozenblat
authors: Physics Department, Bar-Ilan University, Ramat Gan 52900, Israel.
authors: Shlomo
authors: S
authors: Havlin