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

Mingao Yuan, Md. Niamul Islam Sium

TL;DR

This paper proves a central limit theorem for the global clustering coefficient of one-dimensional random geometric graphs with both uniform and nonuniform vertex densities. By deriving precise leading-term expansions for key subgraph counts and exploiting regime-specific techniques (Lyapunov CLT, $U$-statistics with size-dependent kernels, and the method of moments), the authors establish that the centered and scaled clustering coefficient converges to the standard normal distribution across dense, intermediate, and sparse regimes. The limiting centering term $\mu_n$ converges to $3/4$, consistent with known results for the uniform case, while the convergence rates differ between nonuniform and uniform graphs in the dense regime. The results provide detailed insights into how spatial structure and density inhomogeneity influence global clustering and its asymptotic fluctuations, with applications to spatial networks and related data sets.

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 $3/4$

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 of one-dimensional random geometric graphs with both uniform and nonuniform vertex densities. By deriving precise leading-term expansions for key subgraph counts and exploiting regime-specific techniques (Lyapunov CLT, -statistics with size-dependent kernels, and the method of moments), the authors establish that the centered and scaled clustering coefficient converges to the standard normal distribution across dense, intermediate, and sparse regimes. The limiting centering term converges to , consistent with known results for the uniform case, while the convergence rates differ between nonuniform and uniform graphs in the dense regime. The results provide detailed insights into how spatial structure and density inhomogeneity influence global clustering and its asymptotic fluctuations, with applications to spatial networks and related data sets.

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 -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
Paper Structure (6 sections, 7 theorems, 121 equations, 1 table)

This paper contains 6 sections, 7 theorems, 121 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of main results
  4. Proof of Proposition \ref{['propmain']}
  5. Some lemmas
  6. Proof of Theorem \ref{['mthm']}

Key Result

Proposition 2.2

Let $A$ be sampled from $\mathcal{G}_{n}(f,r_n)$. Suppose $f(x)=g(x)I[0\leq x\leq 1]$, where $g(x)$ is a periodic function with period one that is bounded away from zero and has a bounded fourth derivative. In addition, we assume $r_n=o(1)$. Let $\mu_n=\frac{\mathbb{E}[A_{12}A_{13}A_{23}]}{\mathbb{E Here, the remainder terms $O(r_n^3)$, $O(r_n^5)$ and $O(r_n^6)$ do not depend on $X_1$, and

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5