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. Robustness of systems composed of interdependent network layers has been extensively studied in recent years. However, the theoretical frameworks developed so far apply only to special models in the limit of infinite sizes. These methods are therefore of little help in practical contexts, given that real interconnected networks have finite size and their structures are generally not compatible with those of graph toy models. Here, we introduce a theoretical method that takes as inputs the adjacency matrices of the layers to draw the entire phase diagram for the interconnected network, without the need of actually simulating any percolation process. We demonstrate that percolation transitions in arbitrary interdependent networks can be understood by decomposing these system 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 continuous percolation transition. Conversely, if the intersection is dominated by the contribution of the remainders, the transition becomes abrupt even in systems of finite size. We provide examples of real systems that have developed interdependent networks sharing a core of "high quality" edges to prevent catastrophic failures.Percolation is among the most studied topics in statistical physics [1]. The model used to mimic percolation processes assumes the existence of an underlying network of arbitrary structure. Regular grids are traditionally considered to model percolation in materials [2,3]. Complex graphs are instead assumed as underlying supports in the analysis of spreading phenomena in social environments [4,5], or in robustness studies of technological and infrastructural systems [6][7][8]. Once the network has been specified, a configuration of the percolation model is generated assuming nodes (or sites) present with probability p. For p = 0, only a disconnected configuration is possible. For p = 1 instead, all nodes are within the same connected cluster. As the occupation probability varies, the network undergoes a structural transition between these two extreme configurations. Although there are special substrates, e.g., one-dimensional lattices, where the percolation transition may be discontinuous, in the majority of the cases, random percolation models give rise to continuous structural changes [9]. This means that the size of the largest cluster in the network, used as a proxy for the connectedness of the system, increases from the non-percolating to the percolating phases in a smooth fashion.The percolation transition may become discontinuous in a slightly different model involving not just a single network, but a system composed of two or more interdependent graphs [10]. This is a very realistic scenario considering that many, if not all, real graphs are "coupled" with ...

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