Robustness of a Tree-like Network of Interdependent Networks

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˝ os-Rényi (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∞, is given by P∞ = p[1 − exp(−kP∞)] n where 1 − p is the initial fraction of removed nodes. Our general result coincides for n = 1 with the known Erd˝ os-Rényi second-order phase transition at a threshold, p = pc, for a single network. For n = 2 the general result for P∞ corresponds to the n = 2 result [Buldyrev et. al., Nature , 464, (2010)]. Here we show for an NON composed of n coupled RR networks each of degree k, that the giant components is given by P∞ = p 1 − {p 1/n P (n−1)/n ∞ [(1 − (P∞/p) 1/n) (k−1)/k − 1] + 1} k n. Similar to the ER NON, for n = 1 the percolation transition at pc, 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 = kmin, that increases with n, below which the system collapses, there is no such analogous kmin for RR NON system. For any k > 2, the RR NON is stable, i.e., pc < 1. This is due to the critical role played by singly connected nodes which exist in ER NON and enhance the cascading failures but do not exist in the RR NON system.