Multiple percolation transitions in a configuration model of network of networks

Recently much attention has been paid to the study of the robustness of interdependent and multiplex networks and, in particular, networks of networks. The robustness of interdependent networks can be evaluated by the size of a mutually connected component when a fraction of nodes have been removed from these networks. Here we characterize the emergence of the mutually connected component in a network of networks in which every node of a network (layer) $\alpha$ is connected with $q_{\alpha}$ randomly chosen replicas in some other networks and is interdependent of these nodes with probability $r$. We find that when the superdegrees $q_{\alpha}$ of different layers in the network of networks are distributed heterogeneously, multiple percolation phase transition can occur, and depending on the value of $r$ these transition are continuous or discontinuous.