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Normal approximation of subgraph counts in the random-connection model

Qingwei Liu, Nicolas Privault

TL;DR

This work studies normal approximation and rates for subgraph counts $N_G$ in a Poisson-based random-connection model using a cumulant-diagram approach. By expressing cumulants as sums over connected partition diagrams and applying the Statulevičius condition, it derives explicit Kolmogorov-distance rates across dilute, sparse, and full regimes, including tree-like subgraphs in the sparse setting. The framework also specializes to random geometric graphs, providing detailed cumulant bounds in dense, thermodynamic, and sparse regimes and establishing moderate deviations and concentration results. Overall, the paper delivers quantitative central limit theorems for a broad class of spatial random graphs, extending previous Erdős–Rényi and geometry-based results to the general random-connection model with $[0,1]$-valued connections.

Abstract

This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by Möbius inversion. Using the Statulevičius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.

Normal approximation of subgraph counts in the random-connection model

TL;DR

This work studies normal approximation and rates for subgraph counts in a Poisson-based random-connection model using a cumulant-diagram approach. By expressing cumulants as sums over connected partition diagrams and applying the Statulevičius condition, it derives explicit Kolmogorov-distance rates across dilute, sparse, and full regimes, including tree-like subgraphs in the sparse setting. The framework also specializes to random geometric graphs, providing detailed cumulant bounds in dense, thermodynamic, and sparse regimes and establishing moderate deviations and concentration results. Overall, the paper delivers quantitative central limit theorems for a broad class of spatial random graphs, extending previous Erdős–Rényi and geometry-based results to the general random-connection model with -valued connections.

Abstract

This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by Möbius inversion. Using the Statulevičius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.
Paper Structure (9 sections, 18 theorems, 68 equations, 14 figures)

This paper contains 9 sections, 18 theorems, 68 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Set partitions and diagram connectivity
  3. Virtual cumulants
  4. Cumulants of multiparameter stochastic integrals
  5. Cumulants of subgraph counts
  6. Asymptotic growth of subgraph count cumulants
  7. Asymptotic normality of subgraph counts
  8. Subgraph counts in random geometric graphs
  9. Appendix

Key Result

Proposition 1.1

Given $r\geq 2$, consider a connected graph $G$ with $r$ vertices, and a bounded measurable process of the form where $v(x,y)$ is a bounded random process $v(x,y)$ independent of the underlying Poisson point process $\Xi$. Then, the $n$-$th$ moment of the multiparameter stochastic integral is given by the summation where we let $x_{k,l}^\rho:=x_\eta$ whenever $(k,l)\in \eta$, for $\rho \in \Pi(

Figures (14)

  • Figure 1: Two examples of partition diagrams with $\eta = \{2,3,5,8,10\}$, $n=10$, $r=4$.
  • Figure 2: Diagram and graphs $G$, $\rho_G$, $\widetilde{\rho}_G$ with $n=5$, $r=4$.
  • Figure 3: Splitting of the partition $\rho$ with $\rho \vee \pi = \{\pi_1\cup\pi_2, \pi_3\cup\pi_4\cup\pi_5\}$ and $n=5$, $r=4$.
  • Figure 4: Connected non-flat partition diagram with $G$ a cycle graph and $n=5$, $r=4$.
  • Figure 5: Example of graph $\eufrak{g}$ and its spanning tree subgraph.
  • ...and 9 more figures

Theorems & Definitions (30)

  • Proposition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 3.1
  • ...and 20 more