Analytical results for bond percolation and k-core sizes on clustered networks

TL;DR

This paper develops an analytical method to study bond percolation, k-core sizes, and giant component sizes in structured random networks with clustering, revealing how clustering influences percolation thresholds.

Contribution

It introduces a generalized Trapman model for networks with arbitrary degree distributions and clustering, providing new insights into percolation behavior in such networks.

Findings

Clustering increases percolation thresholds in these networks.

High clustering can induce a non-zero epidemic threshold in scale-free networks.

Analytical results contrast with previous studies on unclustered networks.

Abstract

An analytical approach to calculating bond percolation thresholds, sizes of $k$-cores, and sizes of giant connected components on structured random networks with non-zero clustering is presented. The networks are generated using a generalization of Trapman's [P. Trapman, Theor. Pop. Biol. {\bf 71}, 160 (2007)] model of cliques embedded in tree-like random graphs. The resulting networks have arbitrary degree distributions and tunable degree-dependent clustering. The effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature. For very high levels of clustering the percolation threshold in these generalized Trapman networks is increased above the value it takes in a randomly-wired (unclustered) network of the same degree distribution. In assortative scale-free networks, where the variance of the degree distribution is infinite, this clustering effect can lead to a non-zero percolation (epidemic) threshold.