Network research and percolation theory have been focused on the properties of a single isolated network that does not interact or depends on other networks. In reality, many real-networks interact with other networks. We present a framework for studying percolation of interacting networks. In interdependent networks, when nodes in one network fail, they cause dependent nodes in other networks to also fail. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of many interdependent networks. I will present exact analytical solutions for the critical threshold and giant component of a network of n interdependent networks. For n=1 we obtain the classical known result for a single isolated network of second order percolation transition. For n>1 cascading failures occur and the transition becomes a first order. Our results for a network of n interdependent networks suggest that the classical percolation theory extensively studied in physics and mathematics in the past 60 years is only a limiting case of n=1 of a more general case of network of networks.

The rich phenomena found in interdependent networks and the insights obtained from the percolation framework developed have led to a better understanding of the effect of dependence links within single isolated networks. A percolation approach for a single network in the presence of random dependence links was developed recently. The results show that cascading failures occur, yielding a first-order transition, and that the percolation threshold of the network significantly increases with an increase in the number of dependence links.

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