Statistical analysis of edges and bredges in configuration model networks

TL;DR

This paper provides an analytical study of bredges in configuration model networks, revealing their statistical properties, degree correlations, and implications for network resilience and attack vulnerability.

Contribution

It introduces a generating function approach to analytically characterize bredges, including their probability and degree distributions, and uncovers degree correlations specific to bredges on the giant component.

Findings

Bredges are more likely to connect high and low degree nodes.

On finite components, all edges are bredges with no degree correlations.

Degree correlations on the giant component are concentrated on bredges.

Abstract

A bredge (bridge-edge) is an edge whose deletion would split the network component on which it resides into two components. Bredges are vulnerable links that play an important role in network collapse processes, which may result from node or link failures, attacks or epidemics. Therefore, the abundance and properties of bredges affect the resilience of the network. We present analytical results for the statistical properties of bredges in configuration model networks. Using a generating function approach based on the cavity method, we calculate the probability $\hat P(e\in{\rm B})$ that a random edge e in a configuration model network with degree distribution P(k) is a bredge (B). We also calculate the joint degree distribution $\hat P(k,k'|{\rm B})$ of the end-nodes of a random bredge. We examine the distinct properties of bredges on the giant component (GC) and on the finite tree components (FC) of the network. On the finite components all the edges are bredges and there are no degree-degree correlations. We calculate the probability $\hat P(e\in{\rm B}|{\rm GC})$ that a random edge on the giant component is a bredge. We also calculate the joint degree distribution $\hat P(k,k'|{\rm B},{\rm GC})$ of the end-nodes of bredges and the joint degree distribution $\hat P(k,k'|{\rm NB},{\rm GC})$ of the end-nodes of non-bredge (NB) edges on the giant component. Surprisingly, it is found that the degrees k and k' of the end-nodes of bredges are correlated, while the degrees of the end-nodes of NB edges are uncorrelated. We thus conclude that all the degree-degree correlations on the giant component are concentrated on the bredges. We calculate the covariance of end-nodes of bredges and show it is negative, namely bredges tend to connect high degree nodes to low degree nodes. The implications of the results are discussed in the context of common attack scenarios and dismantling processes.