Percolation in real interdependent networks

Abstract

The function of a real network depends not only on the reliability of its own components, but is affected also by the simultaneous operation of other real networks coupled with it. Whereas theoretical methods of direct applicability to real isolated networks exist, the frameworks developed so far in percolation theory for interdependent network layers are of little help in practical contexts, as they are suited only for special models in the limit of infinite size. Here, we introduce a set of heuristic equations that takes as inputs the adjacency matrices of the layers to draw the entire phase diagram for the interconnected network. We demonstrate that percolation transitions in interdependent networks can be understood by decomposing these systems into uncoupled graphs: the intersection among the layers, and the remainders of the layers. When the intersection dominates the remainders, an interconnected network undergoes a smooth percolation transition. Conversely, if the intersection is dominated by the contribution of the remainders, the transition becomes abrupt even in small networks. We provide examples of real systems that have developed interdependent networks sharing cores of ‘high quality’ edges to prevent catastrophic failures. Our understanding of how catastrophe propagates in multi-layered networks relies on theories that apply only to infinite systems. Reducing the interconnected networks to a set of decoupled graphs provides a route to probing finite sizes.