Robustness of a Network of Networks

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 $n$ Erdős-Rényi networks each of average degree $k$, the giant component, $P_{\infty }$, is given by $P_{\infty }=p[1-\exp (-k{P}_{\infty }){]}^n$ where $1-p$ is the initial fraction of removed nodes. Our general result coincides for $n=1$ with the known Erdős-Rényi second-order phase transition for a single network. For any $n\ge 2$ 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.

7 pages, 3 figures

References in corpus (1)

• The critical effect of dependency groups on the function of networks

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