Robustness of a Network of Networks

Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of $n$ interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of $n$ fully dependent Erd\H{o}s-R\'{e}nyi (ER) networks, each of average degree $\bar{k}$, we find that the giant component $P_{\infty}=p[1-\exp(-\bar{k}P_{\infty})]^n$ where $1 - p$ is the initial fraction of removed nodes. This general result coincides for $n = 1$ with the known second-order phase transition for a single network. For any $n>1$ cascading failures occur and the percolation becomes an abrupt first-order transition. (ii) For a starlike network of n partially interdependent ER networks, $P_{\infty}$ depends also on the topology--in contrast to case (i). (iii) For a looplike network formed by $n$ partially dependent ER networks, $P_{\infty}$ is independent of $n$.