Closure coefficients in scale-free complex networks

TL;DR

This paper explores the behavior of the local closure coefficient in scale-free networks, revealing significant differences from multigraph models and relating it to other network metrics.

Contribution

It provides the first exploratory analysis of the local closure coefficient in simple scale-free network models, highlighting its distinct behavior and relationships to other network properties.

Findings

Closure coefficient differs significantly from multigraph models.

High-degree vertices' closure coefficient relates to clustering and nearest neighbor degree.

Analysis focuses on hidden-variable and hyperbolic random graph models.

Abstract

The formation of triangles in complex networks is an important network property that has received tremendous attention. The formation of triangles is often studied through the clustering coefficient. The closure coefficient or transitivity is another method to measure triadic closure. This statistic measures clustering from the head node of a triangle (instead of from the center node, as in the often studied clustering coefficient). We perform a first exploratory analysis of the behavior of the local closure coefficient in two random graph models that create simple networks with power-law degrees: the hidden-variable model and the hyperbolic random graph. We show that the closure coefficient behaves significantly different in these simple random graph models than in the previously studied multigraph models. We also relate the closure coefficient of high-degree vertices to the clustering coefficient and the average nearest neighbor degree.