Number of cliques in random scale-free network ensembles

TL;DR

This paper calculates the average number of cliques in random scale-free networks, showing that unlike Erdős-Rényi graphs, cliques appear even with finite average degree and the largest clique size grows with system size.

Contribution

It provides analytical results for clique counts in scale-free networks using hidden variable and Molloy-Reed ensembles, highlighting differences from Erdős-Rényi graphs.

Findings

Cliques appear in scale-free networks with finite average degree.

Largest clique size diverges as system size increases.

Contrasts with Erdős-Rényi graphs requiring diverging degree for large cliques.

Abstract

In this paper we calculate the average number of cliques in random scale-free networks. 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 Erd\"os 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.