Clustering of random scale-free networks

TL;DR

This paper analyzes how the clustering coefficient in scale-free networks generated by the configuration model depends on network size and degree distribution, revealing size-independent clustering and non self-averaging properties for certain parameters.

Contribution

It derives the finite size dependence of clustering in scale-free networks and uncovers the non self-averaging nature of clustering near degree exponent 2.

Findings

Degree heterogeneity increases clustering levels.

Clustering becomes size independent near gamma=2.

Clustering is non self-averaging for gamma close to 2.

Abstract

We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent $2<\gamma<3$. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of $\gamma \approx 2$, clustering is virtually size independent and, at the same time, becomes a {\it de facto} non self-averaging topological property. This implies that a single instance network is not representative of the ensemble even for very large network sizes.