Percolation of Interdependent Networks with Inter-similarity

TL;DR

This paper develops an analytical framework to understand how inter-similarity in interdependent networks affects cascading failures, revealing that increased inter-similarity reduces cascades but does not change the discontinuous nature of the phase transition.

Contribution

It introduces a percolation model for interdependent networks with inter-similarity, providing analytical solutions and insights into the impact of common links on cascade dynamics.

Findings

Inter-similarity reduces cascade size in interdependent networks.

The phase transition remains discontinuous despite increased inter-similarity.

Analytical solutions are derived for Erdős-Rényi network models.

Abstract

Real data show that interdependent networks usually involve inter-similarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent (Parshani et al \cite{PAR10B}). For example, the coupled world wide 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 another we call these links common links. The fraction of common links in the system is a measure of intersimilarity. Previous simulation results suggest that intersimilarity has considerable effect on reducing the cascading failures, however, a theoretical understanding on this effect on the cascading process is currently missing. Here, we map the cascading process with inter-similarity to a percolation of networks composed of components of common links and non common links. This transforms the percolation of inter-similar 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 Erd\H{o}s-R\'{e}nyi (ER) network with the average degree $K$, and the two networks of non-common links are also ER networks. We show for a fully coupled pair of ER networks, that for any $K\geq0$, 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 networks system.