Open Access
February 2020 Cliques in rank-1 random graphs: The role of inhomogeneity
Kay Bogerd, Rui M. Castro, Remco van der Hofstad
Bernoulli 26(1): 253-285 (February 2020). DOI: 10.3150/19-BEJ1125


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.


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Kay Bogerd. Rui M. Castro. Remco van der Hofstad. "Cliques in rank-1 random graphs: The role of inhomogeneity." Bernoulli 26 (1) 253 - 285, February 2020.


Received: 1 May 2018; Revised: 1 March 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140499
MathSciNet: MR4036034
Digital Object Identifier: 10.3150/19-BEJ1125

Keywords: Clique number , Erdős–Rényi random graphs , inhomogeneous random graphs

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 1 • February 2020
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