Triadic Closure in Configuration Models with Unbounded Degree Fluctuations

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 τ k - 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.