The Fate of Articulation Points and Bredges in Percolation

TL;DR

This paper studies how the probabilities of nodes and edges being articulation points or bredges evolve during random bond removal in networks, using message-passing methods to analyze their distributions.

Contribution

It introduces a message-passing approach to compute the full distributions of articulation point and bredge probabilities in complex networks under percolation.

Findings

Derived distributions for articulation points and bredges in large networks.

Provided closed-form expressions for Erdős-Rényi networks at high mean degree.

Analyzed heterogeneity effects on network dismantling probabilities.

Abstract

We investigate the statistics of articulation points and bredges (bridge-edges) in complex networks in which bonds are randomly removed in a percolation process. Articulation points are nodes in a network which, if removed, would split the network component on which they are located into two or more separate components, while bredges are edges whose removal would split the network component on which they are located into two separate components. Both articulation points and bredges play an important role in processes of network dismantling and it is therefore useful to know the evolution of the probability of nodes or edges to be articulation points and bredges, respectively, when a fraction of edges is randomly removed from the network in a percolation process. Due to the heterogeneity of the network, the probability of a node to be an articulation point, or the probability of an edge to be a bredge will not be homogeneous across the network. We therefore analyze full distributions of articulation point probabilities as well as bredge probabilities, using a message-passing or cavity approach to the problem, as well as a deconvolution of these distributions according to degrees of the node or the degrees of both adjacent nodes in the case of bredges. Our methods allow us to obtain these distributions both for large single instances of networks as well as for ensembles of networks in the configuration model class in the thermodynamic limit of infinite system size. We also derive closed form expressions for the large mean degree limit of Erd\H{o}s-R\'enyi networks.