Percolation in networks of networks with random matching of nodes in different layers

We consider robustness and percolation properties of the networks of networks, in which random nodes in different individual networks (layers) can be interdependent. We explore the emergence of the giant mutually connected component, generalizing the percolation cluster in a single network to interdependent networks, and observe the strong effect of loops of interdependencies. In particular, we find that the giant mutual component does not emerge in a loop formed by any number of layers. In contrast, we observe multiple hybrid transitions in networks of networks formed by infinite number of randomly connected layers, corresponding to the percolation of layers with different number of interdependencies. In particular we find that layers with many interdependencies are more fragile than layers with less interdependencies. These hybrid transitions, combining a discontinuity and a singularity, are responsible for joining a finite fraction of nodes in different layers to the giant mutually connected component. In the case of partial interdependence, when only a fraction of interlinks between layers provide interdependence, some of these transitions can become continuous.