Global clustering coefficient in scale-free networks

TL;DR

This paper investigates the global clustering coefficient in scale-free networks with infinite variance degree distributions, proving it tends to zero as network size increases, contrasting with the behavior of local clustering.

Contribution

It provides a theoretical proof that in scale-free networks with infinite variance degree distributions, the global clustering coefficient diminishes to zero as the network grows.

Findings

Global clustering coefficient tends to zero in infinite variance scale-free networks.

Contrasts with models where local clustering remains positive.

Provides theoretical insight into clustering behavior in real-world networks.

Abstract

In this paper, we analyze the behavior of the global clustering coefficient in scale free graphs. We are especially interested in the case of degree distribution with an infinite variance, since such degree distribution is usually observed in real-world networks of diverse nature. There are two common definitions of the clustering coefficient of a graph: global clustering and average local clustering. It is widely believed that in real networks both clustering coefficients tend to some positive constant as the networks grow. There are several models for which the average local clustering coefficient tends to a positive constant. On the other hand, there are no models of scale-free networks with an infinite variance of degree distribution and with a constant global clustering. In this paper we prove that if the degree distribution obeys the power law with an infinite variance, then the global clustering coefficient tends to zero with high probability as the size of a graph grows.