Percolation of a general network of networks

TL;DR

This paper develops an analytical framework to study the percolation properties of interdependent networks, revealing how cascading failures can cause abrupt system collapses and identifying conditions for different phase transitions.

Contribution

It introduces a general analytical framework for interdependent networks, analyzing percolation transitions and the effects of network topology and feedback on system robustness.

Findings

Interdependent networks can exhibit discontinuous percolation transitions due to cascading failures.

Loops in the network topology influence the size of the largest connected component.

Feedback coupling can lead to only second-order transitions or abrupt collapses.

Abstract

Percolation theory is an approach to study vulnerability of a system. We develop analytical framework and analyze 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 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 abrupt collapse of the system. We demonstrate our general framework for a NetONet composed of $n$ classic Erd\H{o}s-R\'{e}nyi (ER) networks, where each network depends on the same number $m$ of other networks, i.e., a random regular network of interdependent ER networks. In contrast to a \emph{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 $m$. We also analyzed the extremely vulnerable feedback condition of coupling. In the case of ER networks, the NetONet only exhibits two phases, a second order phase transition and collapse, and there is no first phase transition regime unlike the no feedback condition. In the case of NetONet composed of RR networks, there exists a first order phase transition when $q$ is large and second order phase transition when $q$ is small. Our results can help in designing robust interdependent systems.