Abstract
As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two cliques means that they share at least 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 of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction of vertices in the giant clique cluster for makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for . More interesting, our analysis shows that at the critical point, the order parameter for is neither 0 nor 1, but a constant depending on and . All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.
- Received 11 August 2015
DOI:https://doi.org/10.1103/PhysRevE.92.042116
©2015 American Physical Society