Cascading failures in networks of networks linked by directional and bidirectional hyperlinks

We study the cascading failures in systems of any number of networks connected to each other via groups of supply links, which we call hyperlinks. The individual supply links of a hyperlink connect individual nodes belonging to the pair of networks connected by this hyperlink. Such a system is called a network of networks (NON). NONs based on the idea of mutual percolation have been studied for the case of dependency hyperlinks. The present model generalizes the heterogeneous k-core percolation for the NON, where any number of hyperlinks can be directional and any number of them can be bidirectional. We show that, by utilizing generating function formalism, the cascading process can be modeled by a set of recursive relations that are generalizations of previously studied relations in heterogeneous k-core percolation for single or bipartite networks. We demonstrate that the order in which failures propagate throughout the system does not matter for determining the final fraction of functional nodes, which depends only on the NON topology. We show that, in the NONs, there can be more than one transition point, defined as the discontinuous jump in the fraction of functional nodes at the end of the cascade as the strength of the initial attack on one of the networks gradually changes, and more than one critical point, defined as when the behavior changes from continuous to discontinuous. We find that the number of these points is strongly related to the number of hyperlinks in the NON. We further generalize previously studied criteria for the transition points and critical points in the bipartite network to the NONs with hyperlinks, and compare the phase diagrams of NONs with multiple critical points to the phase diagrams of protein solutions.