Global clustering coefficient in scale-free weighted and unweighted networks

TL;DR

This paper investigates the behavior of the global clustering coefficient in scale-free networks, showing it tends to zero in unweighted graphs with infinite variance but can remain constant in weighted graphs.

Contribution

It provides a theoretical analysis of the global clustering coefficient in scale-free networks with infinite variance, highlighting differences between weighted and unweighted cases.

Findings

Global clustering coefficient tends to zero in unweighted graphs with infinite variance.

Weighted graphs can maintain a constant global clustering coefficient despite infinite variance.

The paper offers bounds on the maximum clustering coefficient in such networks.

Abstract

In this paper, we present a detailed analysis of the global clustering coefficient in scale-free graphs. Many observed real-world networks of diverse nature have a power-law degree distribution. Moreover, the observed degree distribution usually has an infinite variance. Therefore, we are especially interested in such degree distributions. In addition, we analyze the clustering coefficient for both weighted and unweighted graphs. There are two well-known definitions of the clustering coefficient of a graph: the global and the average local clustering coefficients. There are several models proposed in the literature for which the average local clustering coefficient tends to a positive constant as a graph grows. On the other hand, there are no models of scale-free networks with an infinite variance of the degree distribution and with an asymptotically constant global clustering coefficient. Models with constant global clustering and finite variance were also proposed. Therefore, in this paper we focus only on the most interesting case: we analyze the global clustering coefficient for graphs with an infinite variance of the degree distribution. For unweighted graphs, we prove that the global clustering coefficient tends to zero with high probability and we also estimate the largest possible clustering coefficient for such graphs. On the contrary, for weighted graphs, the constant global clustering coefficient can be obtained even for the case of an infinite variance of the degree distribution.