Clique percolation in random graphs

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.