Disparity of clustering coefficients in the Holme-Kim network model

TL;DR

This paper investigates how the Holme-Kim network model's clustering coefficients behave, revealing that local clustering remains positive while global clustering diminishes slowly, depending on the definition used.

Contribution

It provides a detailed analysis of clustering coefficient disparities in the Holme-Kim model using martingale techniques, highlighting the dependence on the clustering definition.

Findings

Local clustering coefficient remains positive

Global clustering coefficient tends to zero slowly

Clustering behavior depends on the coefficient definition

Abstract

The Holme-Kim random graph processes is a variant of the Barabasi-Albert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that local clustering coefficient remains typically positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.