General clique percolation in random networks

Abstract

A general clique community of a network, which consists of adjacent k-cliques sharing at least l vertices with , is introduced. With the emergence of a giant clique community in the network, there is a clique percolation. Using the largest size jump Δ of the largest clique community during network evolution and the corresponding evolution step Tc, we study the general clique percolation of the Erdős-Rényi network. We investigate the averages of Δ and Tc and their fluctuations for different network size N. The clique percolation can be identified by the power-law finite-size effects of the averages and root mean squares of fluctuation. The finite-size scaling distribution functions of fluctuations are calculated. The universality class of the clique percolation is characterized by the critical exponents of power-law finite-size effects. Using Monte Carlo simulations, we find that the Erdős-Rényi network experiences a series of clique percolation with . We find that the critical exponents and therefore the universality class of the clique percolation depend on clique connection index l, but are independent of clique size k.