Percolation of interdependent network of networks

Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many real-world networks interact with and depend on each other. Very recently an analytical framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a Network Of Networks (NONs) formed by n interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (first-order) phase transition, unlike the well-known continuous second-order transition in single isolated networks. Moreover, although networks with broader degree distributions, e.g., scale-free networks, are more robust when analyzed as single networks, they become more vulnerable in a NON. We also review the effect of space embedding on network vulnerability. It is shown that for spatially embedded networks any finite fraction of dependency nodes will lead to abrupt transition