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 (), i.e., the probability that two neighbors of a degree- node are neighbors themselves. We show that () progressively falls off with and the graph size and eventually for settles on a power law with 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.