Robustness of a Network of Networks

TL;DR

This paper develops an analytical framework for studying the robustness of interconnected networks, revealing that interdependence causes a shift from second-order to first-order percolation transitions, with implications for network resilience.

Contribution

It introduces an exact percolation law for a network of interdependent Erd ext{o}s-Rényi networks, extending classical percolation theory to account for network interactions.

Findings

Interdependent networks exhibit cascading failures not seen in isolated networks.

The transition in network robustness shifts from second-order to first-order with multiple networks.

The derived law generalizes classical percolation results to interconnected systems.

Abstract

Almost all network research has been focused on the properties of a single network that does not interact and depends on other networks. In reality, many real-world networks interact with other networks. Here we develop an analytical framework for studying interacting networks and present an exact percolation law for a network of $n$ interdependent networks. In particular, we find that for $n$ Erd\H{o}s-R\'{e}nyi networks each of average degree $k$, the giant component, $P_{\infty}$, is given by $P_{\infty}=p[1-\exp(-kP_{\infty})]^n$ where $1-p$ is the initial fraction of removed nodes. Our general result coincides for $n=1$ with the known Erd\H{o}s-R\'{e}nyi second-order phase transition for a single network. For any $n \geq 2$ cascading failures occur and the transition becomes a first-order percolation transition. The new law for $P_{\infty}$ shows that percolation theory that is extensively studied in physics and mathematics is a limiting case ($n=1$) of a more general general and different percolation law for interdependent networks.