Emergence of large cliques in random scale-free network

TL;DR

This paper investigates the emergence of large cliques in uncorrelated scale-free networks, revealing that such networks can have large, diverging cliques even with finite average degree, especially when the degree exponent is less than 3.

Contribution

It provides bounds on the clique number in scale-free networks and shows large cliques can exist with finite average degree for certain degree exponents.

Findings

Large cliques appear in scale-free networks with finite average degree.

Maximal clique size diverges with system size for gamma<3.

Large cliques are present even in networks with power-law degree distributions.

Abstract

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. Interesting 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 gamma in the interval (2,3). Moreover as long as gamma<3 scale-free networks have a maximal clique which diverges with the system size.