Central limit theorem for the global clustering coefficient of random geometric graphs

TL;DR

This paper proves a central limit theorem for the global clustering coefficient in random geometric graphs, revealing its asymptotic normality and convergence rates across different regimes, with the coefficient approaching 3/4.

Contribution

It establishes the first CLT for the global clustering coefficient in RGGs, detailing the asymptotic distribution and convergence behavior in various regimes.

Findings

Global clustering coefficient converges to 3/4 asymptotically.

Different convergence rates in dense and sparse regimes.

Established CLT using Lyapunov, U-statistics, and method of moments.

Abstract

The global clustering coefficient serves as a powerful metric for the structural analysis and comparison of complex networks. Random geometric graphs offer a realistic framework for representing the spatial constraints and geometry often found in real-world network datasets. In this paper, we establish a central limit theorem for the global clustering coefficient of random geometric graphs. Our main result identifies the centering and scaling sequences required for convergence in law to the standard normal distribution. Our approach varies by regime: in the dense case, we employ the Lyapunov CLT; in the intermediate case, we utilize the asymptotic theory of $U$-statistics with sample-size-dependent kernels; and in the sparse regime, we use the method of moments to derive the asymptotic distribution. Notably, the convergence rates for non-uniform and uniform random geometric graphs diverge in the dense regime, yet they coincide in the sparse regime. In addition, we find that the global clustering coefficient for both uniform and non-uniform RGGs is asymptotically equal to $3/4$