Clique percolation in random graphs

TL;DR

This paper provides a theoretical analysis of clique percolation in Erdős-Rényi graphs, revealing the nature of phase transitions for different parameters and confirming previous simulation results.

Contribution

It develops an exact theoretical framework for clique percolation, including critical points and order parameters, and characterizes the phase transition types for various clique connection scenarios.

Findings

Fraction of cliques in the giant cluster undergoes continuous transition.

Fraction of vertices in the giant cluster exhibits step-function-like discontinuous transition for l>1.

Order parameter at criticality is a constant depending on k and l.

Abstract

As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two $k$-cliques means that they share at least $l<k$ vertices. In this paper, we develop a theoretical approach to study clique percolation in Erd\H{o}s-R\'{e}nyi graphs, which gives not only the exact solutions of the critical point, but also the corresponding order parameter. Based on this, we prove theoretically that the fraction $\psi$ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction $\phi$ of vertices in the giant clique cluster for $l>1$ makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for $l=1$. More interesting, our analysis shows that at the critical point, the order parameter $\phi_c$ for $l>1$ is neither $0$ nor $1$, but a constant depending on $k$ and $l$. All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.