Number of cliques in random scale-free network ensembles

In this paper we calculate the average number of cliques in random scale-free networks in the limit of large network size N ≫ 1 . We consider first the hidden variable ensemble and subsequently the Molloy Reed ensemble. In both cases we find that cliques, i.e. fully connected subgraphs, appear also when the average degree is finite. This is in contrast to what happens in Erdös and Renyi graphs in which diverging average degree is required to observe cliques of size c > 3 . Moreover we show that in random scale-free networks the clique number, i.e. the size of the largest clique present in the network, diverges with the system size.