title: Clique percolation in random graphs.
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 Erdős-Rényi 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 ψ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction ϕ 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 ϕ(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.
authors.affiliation: Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China.
authors.forename: Ming
authors.initials: M
authors.lastname: Li
authors.affiliation: Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China.|Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, People's Republic of China.
authors.forename: Youjin
authors.initials: Y
authors.lastname: Deng
authors.affiliation: Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China.
authors.forename: Bing-Hong
authors.initials: BH
authors.lastname: Wang