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

Research output: Contribution to journalArticleAcademicpeer-review

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

Cite this

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title = "Cliques in rank-1 random graphs: The role of inhomogeneity",
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.",
keywords = "clique number, Erdos-Renyi random graphs, inhomogeneous random graphs, Erdos-R{\'e}nyi random graphs, Clique number, Inhomogeneous random graphs",
author = "Kay Bogerd and Castro, {Rui M.} and {van Der Hofstad}, Remco",
year = "2020",
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language = "English",
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Cliques in rank-1 random graphs : The role of inhomogeneity. / Bogerd, Kay (Corresponding author); Castro, Rui M.; van Der Hofstad, Remco.

In: Bernoulli, Vol. 26, No. 1, 02.2020, p. 253-285.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Cliques in rank-1 random graphs

T2 - The role of inhomogeneity

AU - Bogerd, Kay

AU - Castro, Rui M.

AU - van Der Hofstad, Remco

PY - 2020/2

Y1 - 2020/2

N2 - 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.

AB - 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.

KW - clique number

KW - Erdos-Renyi random graphs

KW - inhomogeneous random graphs

KW - Erdos-Rényi random graphs

KW - Clique number

KW - Inhomogeneous random graphs

UR - https://arxiv.org/abs/1805.01688

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