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Preprint

Jianxi Gao , Sergey V. Buldyrev , Shlomo Havlin , H. Eugene Stanley

27 October 2010

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.

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D. Watts, S Strogatz, M. Newman (2000)

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D. Watts, S Strogatz, M. Newman … (2000)

10.1103/PhysRevLett.107.195701

1010.5829

http://creativecommons.org/licenses/by/3.0/