Emergence of large cliques in random scale-free networks

In a network cliques are fully connected subgraphs that reveal which are the tight communities present in it. Cliques of size c > 3 are present in random Erdos and Renyi graphs only in the limit of diverging average connectivity. Starting from the finding that real scale-free graphs have large cliques, we study the clique number in uncorrelated scale-free networks finding both upper and lower bounds. Interestingly, we find that in scale-free networks large cliques appear also when the average degree is finite, i.e. even for networks with power law degree distribution exponents γ(2,3). Moreover, as long as γ < 3, scale-free networks have a maximal clique which diverges with the system size.

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