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

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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 loglog(n) multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.
Original languageEnglish
Pages (from-to)253-285
Number of pages33
JournalBernoulli
Volume26
Issue number1
DOIs
Publication statusPublished - Feb 2020

Keywords

  • clique number
  • Erdos-Renyi random graphs
  • inhomogeneous random graphs
  • Erdos-Rényi random graphs
  • Clique number
  • Inhomogeneous random graphs

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