Random graphs with arbitrary clustering and their applications

TL;DR

This paper extends analytical models of random graphs to include arbitrary clustering and multilayer networks, enabling better understanding of their percolation properties beyond traditional tree-like assumptions.

Contribution

It introduces a generalized model for random graphs with arbitrary clustering, including multilayer networks, expanding the applicability of existing percolation analysis methods.

Findings

Analytical solutions for percolation in networks with arbitrary clustering.

Extension of the configuration model to multilayer and non-degree-equivalent clusters.

Numerical examples demonstrating the model's applicability to complex networks.

Abstract

The structure of many real networks is not locally tree-like and hence, network analysis fails to characterise their bond percolation properties. In a recent paper [P. Mann, V. A. Smith, J. B. O. Mitchell, and S. Dobson, Percolation in random graphs with higher-order clustering, arXiv e-prints, p. arXiv:2006.06744, June 2020.], we developed analytical solutions to the percolation properties of random networks with homogeneous clustering (clusters whose nodes are degree-equivalent). In this paper, we extend this model to investigate networks that contain clusters whose nodes are not degree-equivalent, including multilayer networks. Through numerical examples we show how this method can be used to investigate the properties of random complex networks with arbitrary clustering, extending the applicability of the configuration model and generating function formulation.