Clique Percolation in Random Networks

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 Erdős-Rényi 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 p_c(k)=[(k-1)N]^-1/(k-1). At the transition point the scaling of the giant component with N is highly nontrivial 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.