Robustness of a Tree-like Network of Interdependent Networks

TL;DR

This paper develops an analytical framework to study the robustness and cascading failures in interdependent networks, revealing that regular networks are more resilient than Erdős-Rényi networks when coupled.

Contribution

It introduces a general exact percolation law for a network of interdependent networks and compares the robustness of coupled ER and RR networks.

Findings

RR networks exhibit higher robustness than ER networks.

First order percolation transition occurs for multiple coupled networks.

No minimum degree threshold for RR networks' stability.

Abstract

In reality, many real-world networks interact with and depend on other networks. We develop an analytical framework for studying interacting networks and present an exact percolation law for a network of $n$ interdependent networks (NON). We present a general framework to study the dynamics of the cascading failures process at each step caused by an initial failure occurring in the NON system. We study and compare both $n$ coupled Erd\H{o}s-R\'{e}nyi (ER) graphs and $n$ coupled random regular (RR) graphs. We found recently [Gao et. al. arXive:1010.5829] that for an NON composed of $n$ ER 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 at a threshold, $p=p_c$, for a single network. For $n=2$ the general result for $P_{\infty}$ corresponds to the $n=2$ result [Buldyrev et. al., Nature, 464, (2010)]. Similar to the ER NON, for $n=1$ the percolation transition at $p_c$, is of second order while for any $n>1$ it is of first order. The first order percolation transition in both ER and RR (for $n>1$) is accompanied by cascading failures between the networks due to their interdependencies. However, we find that the robustness of $n$ coupled RR networks of degree $k$ is dramatically higher compared to the $n$ coupled ER networks of average degree $k$. While for ER NON there exists a critical minimum average degree $k=k_{\min}$, that increases with $n$, below which the system collapses, there is no such analogous $k_{\min}$ for RR NON system.