Abstract

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. As I will show, this general theory has many novel features that are not present in classical percolation theory. For example, analyzing complex systems as a set of interdependent networks may alter a basic assumption that network theory has relied on: while for a single network a broader degree distribution of the network nodes results in the network being more robust to random failures, for interdependent networks, the broader the distribution is, the more vulnerable the networks become to random failure. We also show that reducing the coupling between the networks leads to a change from a first order percolation phase transition to a second order percolation transition at a critical point.

References:
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