Percolation in random graphs with higher-order clustering

TL;DR

This paper extends percolation theory to clustered networks with cycles and cliques, deriving analytical criteria for the emergence of giant components and analyzing how clustering affects network connectivity.

Contribution

It generalizes the Molloy-Reed criterion to networks with higher-order clustering, providing analytical solutions for the size of the giant component in such networks.

Findings

Larger simple cycles make networks more tree-like.

Clustering with larger cliques deviates from tree-like behavior.

Degree-assortativity influences the impact of clustering on percolation.

Abstract

Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally tree-like and does not contain short-range loops between neighbours. In this paper we use the generating function formulation to examine clustered networks that contain simple cycles and cliques of any order. We use the natural generalisation to the Molloy-Reed criterion for these networks to describe their critical properties and derive an analytical description of the size of the giant component, providing solutions for Poisson and power-law networks. We find that networks comprising larger simple cycles behave increasingly more tree-like. Conversley, clustering comprised of larger cliques increasingly deviate from the tree-like solution, although the behaviour is strongly dependent on the degree-assortativity.