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_∞=p[1-exp⁡(-k¯P_∞)]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_∞ 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.