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

2026-02-19

AI Summary

This study investigates the asymptotic distribution of the global clustering coefficient in random geometric graphs across three density regimes: dense, intermediate, and sparse. By leveraging the Lyapunov central limit theorem, theory of kernel U-statistics with sample-dependent kernels, and moment methods, the authors systematically derive the centering and scaling sequences under which the coefficient converges to a standard normal distribution in each regime. The main contributions include the first complete characterization of central limit theorems for the global clustering coefficient in both homogeneous and inhomogeneous models, a detailed comparison of convergence rates between dense and sparse regimes, and the proof that the asymptotic mean is universally equal to 3/4 regardless of the underlying density regime.

Technology Category

Machine Learning: ClusteringReasoning under Uncertainty: Graphical ModelsData Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web textSecurity and Privacy: Large-scale security measurements

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 $3/4$

Problem

• central limit theorem
• global clustering coefficient
• random geometric graphs
• asymptotic distribution
• spatial networks

Innovation

• central limit theorem
• random geometric graphs
• global clustering coefficient
• U-statistics
• asymptotic distribution