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 . of the largest clique community during network evolution and the corresponding evolution step Tc, we study the general (k, l) clique percolation of the Erd.os-Rényi network. We investigate the averages of . 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 (k, l) clique percolation is characterized by the critical exponents of power-law finite-size effects. Using Monte Carlo simulations, we find that the Erd.os-Rényi 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.