Percolation of random clique networks

Abstract

We propose a random clique network (RCN), which is constructed by adding cliques randomly. In a k -clique, there are k nodes which are connected each other completely. The RCN possesses some characters of the small world network and the modular hierarchical structure. At k =2, the RCN becomes the Erdos-Renyi (ER) random network. In this paper, we study the percolation transition of RCN by investigating the biggest size gap Δ of the largest cluster in the network and the corresponding evolution step, which is taken as the transition point. From the Monte Carlo simulations of RCN at k =2, 3, 4, 5, we can calculate the mean values and the mean square root of fluctuations for Δ and the transition point. They all show a power-law dependence on the network size N . This leads to the conclusion that the percolation transitions of RCN at k =2, 3, 4, 5 are continuous. From the exponents of power-law behaviors, the critical exponents β 1, ν 1 of Δ and the critical exponents β 2, ν 2 of the transition points can be obtained. These critical exponents of different RCN are shown to be independent of the clique size k . The percolation transitions of RCN belong to the same universality class as the ER random network.