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We consider robustness and percolation properties of the networks ofnetworks, in which random nodes in different individual networks (layers) canbe interdependent. We explore the emergence of the giant mutually connectedcomponent, generalizing the percolation cluster in a single network tointerdependent networks, and observe the strong effect of loops ofinterdependencies. In particular, we find that the giant mutual component doesnot emerge in a loop formed by any number of layers. In contrast, we observemultiple hybrid transitions in networks of networks formed by infinite numberof randomly connected layers, corresponding to the percolation of layers withdifferent number of interdependencies. In particular we find that layers withmany interdependencies are more fragile than layers with lessinterdependencies. These hybrid transitions, combining a discontinuity and asingularity, are responsible for joining a finite fraction of nodes indifferent layers to the giant mutually connected component. In the case ofpartial interdependence, when only a fraction of interlinks between layersprovide interdependence, some of these transitions can become continuous.

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