Degree and clustering coefficient in sparse random intersection graphs

TL;DR

This paper rigorously analyzes the degree distribution and clustering coefficient in sparse random intersection graphs, explaining empirical observations about their negative correlation and $k^{-1}$ scaling behavior.

Contribution

It provides mathematically rigorous results linking degree distribution and clustering coefficient in two models, clarifying empirical phenomena observed in real networks.

Findings

Clustering coefficient negatively correlates with degree.

Local clustering coefficient scales as $k^{-1}$ with degree k.

Results are consistent with empirical data from web graphs and biological networks.

Abstract

We establish asymptotic vertex degree distribution and examine its relation to the clustering coefficient in two popular random intersection graph models of Godehardt and Jaworski [Electron. Notes Discrete Math. 10 (2001) 129-132]. For sparse graphs with a positive clustering coefficient, we examine statistical dependence between the (local) clustering coefficient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of Foudalis et al. [In Algorithms and Models for Web Graph (2011) Springer] that, "clustering correlates negatively with degree." Moreover, they explain empirical results on $k^{-1}$ scaling of the local clustering coefficient of a vertex of degree k reported in Ravasz and Barabasi [Phys. Rev. E 67 (2003) 026112].