The critical point of k-clique percolation in the Erdos-Renyi graph

TL;DR

This paper analyzes the phase transition in k-clique percolation within Erdős-Rényi graphs, deriving an explicit critical probability for the emergence of a giant k-clique cluster.

Contribution

It provides an analytical expression for the critical linking probability p_c(k) in Erdős-Rényi graphs, enhancing understanding of community detection thresholds.

Findings

Derived the critical probability p_c(k) for k-clique percolation

Established the relation between average cluster size and percolation threshold

Provided a generating function approach to analyze percolation behavior

Abstract

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdos-Renyi graph. When the probability p of two nodes being connected is above a certain threshold p_c(k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p_c(k).