Asymptotic normality for triangle counting in the sparse $\beta$-model

TL;DR

This paper establishes the asymptotic normality of triangle counts in the sparse $eta$-model, a network model capturing degree heterogeneity, using advanced probabilistic methods to analyze their distribution as the network size grows.

Contribution

It provides the first rigorous proof of asymptotic normality for triangle counts in the sparse $eta$-model, including explicit bounds and conditions based on degree heterogeneity.

Findings

Derived the asymptotic mean and variance of triangle counts.

Established a non-asymptotic bound on the distributional distance to normality.

Proved asymptotic normality under certain degree heterogeneity conditions.

Abstract

We study the number of triangles $T_n$ in the sparse $\beta$-model on $n$ vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of $T_n$. Next, by applying the Malliavin-Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between normalized $T_n$ and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for $T_n$, as $n\to\infty$.