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 Erdős-Rényi (ER) networks, each of average degree k, we find that the giant component is P∞ =p1-exp(-kP∞) 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∞ depends also on the topology-in contrast to case (i). (iii) For a looplike network formed by n partially dependent ER networks, P∞ is independent of n.