Clique percolation in random networks

TL;DR

This paper introduces the concept of k-clique percolation in random graphs, analyzing the conditions for giant component formation and highlighting its potential for identifying overlapping communities in large networks.

Contribution

It provides an analytical and numerical study of k-clique percolation thresholds and scaling behavior in Erdős-Rényi graphs, proposing a novel method for community detection.

Findings

Percolation transition occurs at a specific edge probability threshold.

Scaling of the giant component depends on the clique size k.

Clique percolation effectively identifies overlapping communities.

Abstract

The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdos-Renyi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold pc(k)=[(k-1)N]^{-1/(k-1)}. At the transition point the scaling of the giant component with N is highly non-trivial and depends on k. We discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks.