Heterogeneity in outcomes of repeated instances of percolation experiments.

Abstract

We investigate the heterogeneity of outcomes of repeated instances of percolation experiments in complex networks using a message-passing approach to evaluate heterogeneous, node-dependent probabilities of belonging to the giant or percolating cluster, i.e., the set of mutually connected nodes whose size scales linearly with the size of the system. We evaluate these both for large finite single instances and for synthetic networks in the configuration model class in the thermodynamic limit. For the latter, we consider both Erdős-Rényi and scale-free networks as examples of networks with narrow and broad degree distributions, respectively. For real-world networks we use an undirected version of a Gnutella peer-to-peer file-sharing network with N=62568 nodes as an example. We derive the theory for multiple instances of both uncorrelated and correlated percolation processes. For the uncorrelated case, we also obtain a closed-form approximation for the large mean degree limit of Erdős-Rényi networks.