Asymptotic distributions of the average clustering coefficient and its variant

TL;DR

This paper studies the asymptotic behavior of the average clustering coefficient and its variant in heterogeneous Erdős-Rényi graphs, revealing a normal distribution convergence and a phase transition in variance.

Contribution

It provides the first asymptotic distributions for these network statistics, highlighting differences and phase transition phenomena in their variances.

Findings

Standardized average clustering coefficient converges to normal distribution.

Variance of the average clustering coefficient exhibits a phase transition.

Sum of weighted triangles does not show a phase transition in variance.

Abstract

In network data analysis, summary statistics of a network can provide us with meaningful insight into the structure of the network. The average clustering coefficient is one of the most popular and widely used network statistics. In this paper, we investigate the asymptotic distributions of the average clustering coefficient and its variant of a heterogeneous Erd\"{o}s-R\'{e}nyi random graph. We show that the standardized average clustering coefficient converges in distribution to the standard normal distribution. Interestingly, the variance of the average clustering coefficient exhibits a phase transition phenomenon. The sum of weighted triangles is a variant of the average clustering coefficient. It is recently introduced to detect geometry in a network. We also derive the asymptotic distribution of the sum weighted triangles, which does not exhibit a phase transition phenomenon as the average clustering coefficient. This result signifies the difference between the two summary statistics.