Cliques in rank-1 random graphs: the role of inhomogeneity

TL;DR

This paper analyzes the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, revealing its concentration properties and how inhomogeneity influences its size.

Contribution

It provides a precise expression for the clique number's concentration and clarifies the impact of inhomogeneity on its order in rank-1 random graphs.

Findings

Clique number concentrates on at most two integers.

Inhomogeneity affects only multiplicative constants or a log-log factor.

For sparse graphs, the clique number is bounded and inhomogeneity effects vanish.

Abstract

We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $\log\log(n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.