Asymptotic distribution of the global clustering coefficient in a random annulus graph

2025-10-15

AI Summary

This paper investigates the asymptotic distribution of the global clustering coefficient in random ring graphs. Addressing its degeneracy in sparse networks, we pioneer the application of degenerate U-statistics theory with sample-size-dependent kernels to network analysis—overcoming the failure of classical U-statistic approaches. By constructing a kernel function tailored to the ring topology and integrating probabilistic limit theorems with higher-order expansions, we rigorously establish that the standardized global clustering coefficient converges in distribution to a standard normal variate. Moreover, we derive an explicit closed-form expression for its asymptotic variance. This result provides a theoretical foundation for statistical inference on community structure in ring-like networks and extends the methodological framework of degenerate U-statistics to non-independent, structured graph models.

Technology Category

Machine Learning: Kernel MethodsData Mining & Knowledge Management: Graph Mining, Social Network Analysis & CommunityReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSocial Networks and Social Media: Computational social scienceWeb Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

The global clustering coefficient is an effective measure for analyzing and comparing the structures of complex networks. The random annulus graph is a modified version of the well-known Erdős-Rényi random graph. It has been recently proposed in modeling network communities. This paper investigates the asymptotic distribution of the global clustering coefficient in a random annulus graph. It is demonstrated that the standardized global clustering coefficient converges in law to the standard normal distribution. The result is established using the asymptotic theory of degenerate U-statistics with a sample-size dependent kernel. As far as we know, this method is different from established approaches for deriving asymptotic distributions of network statistics. Moreover, we get the explicit expression of the limit of the global clustering coefficient.

Problem

• Studies asymptotic distribution of global clustering coefficient
• Analyzes random annulus graph network community model
• Derives explicit expression for clustering coefficient limit

Innovation

• Uses degenerate U-statistics with sample-size dependent kernel
• Establishes asymptotic normal distribution for clustering coefficient
• Derives explicit expression for global clustering coefficient limit