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 Erdős-Rényi 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 Erdős-Rényi second-order phase transition for a single network. For any n ≥ 2 cascading failures occur and the transition becomes a first-order percolation transition. The new law for P∞ 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.
Gao et al. REF proposed a framework for studying the percolation of n interdependent networks.
Robustness of a Network of Networks
Phys. Rev. Lett. 107, 195701 (2011)
Medicine Physics Computer Science