Triadic closure in configuration models with unbounded degree fluctuations

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The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k), i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and the graph size n and eventually for k=Ω(n) settles on a power law c(k) ∼ n 5 - 2 τk - 2 ( 3 - τ ) with τ∈ (2 , 3) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

Original languageEnglish
Pages (from-to)746-774
Number of pages29
JournalJournal of Statistical Physics
Issue number3-4
Publication statusPublished - 1 Nov 2018


  • Clustering
  • Configuration model
  • Random graphs


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