Optimal subgraph structures in scale-free configuration models

Remco Van der Hofstad, Johan S.H. Van Leeuwaarden, Clara Stegehuis

Research output: Contribution to journalArticleAcademicpeer-review

6 Citations (Scopus)
90 Downloads (Pure)


Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. We obtain these results by analyzing the configuration model with degree exponent τ ∈ (2, 3) and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize all subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the erased configuration model.

Original languageEnglish
Pages (from-to)501-537
Number of pages37
JournalAnnals of applied probability
Issue number2
Publication statusPublished - Apr 2021


  • Configuration model
  • Motifs
  • Random graphs
  • Subgraphs


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