Triadic closure in configuration models with unbounded degree fluctuations

TL;DR

This paper analyzes the local clustering coefficient in configuration models with unbounded degree fluctuations, revealing a power-law decay with degree and providing insights into the structure of scale-free networks.

Contribution

It characterizes the behavior of local clustering in configuration models with power-law degrees, extending understanding of network modularity and hierarchical structure.

Findings

Clustering coefficient c(k) decreases with degree k

For large k, c(k) follows a power law c(k) ~ k^{-2(3-τ)}

Results align with observations in real-world networks

Abstract

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 eventually for $k=\Omega(\sqrt{n})$ settles on a power law $c(k)\sim k^{-2(3-\tau)}$ with $\tau\in(2,3)$ 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.