Asymptotics for Cliques in Scale-Free Random Graphs

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In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung–Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities h1−αl(h), where α>2 and l is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all non-integer α>2. We also explain why the case of an integer α is different, and present partial results for the asymptotics in that case.
Original languageEnglish
Article number19
JournalJournal of Statistical Physics
Issue number2
Early online date3 Sept 2022
Publication statusPublished - Nov 2022


  • Chung–Lu model
  • Cliques
  • Inhomogeneous random graph
  • Scale–free networks
  • Slowly varying function

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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