General clique percolation in random networks

A general (k, l) clique community of a network, which consists of adjacent k- cliques sharing at least l vertices with k - 1 >= l >= 1, is introduced. With the emergence of a giant (k, l) clique community in the network, there is a (k, l) clique percolation. Using the largest size jump Delta of the largest clique community during network evolution and the corresponding evolution step T-c, we study the general (k, l) clique percolation of the Erdos-Renyi network. We investigate the averages of Delta and T-c 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 (k, l) clique percolation is characterized by the critical exponents of power-law finite-size effects. Using Monte Carlo simulations, we find that the Erdos-Renyi network experiences a series of (k, l) clique percolation with (k, l) = (2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1). We find that the critical exponents and therefore the universality class of the (k, l) clique percolation depend on clique connection index l, but are independent of clique size k. Copyright (C) EPLA, 2014