Normal approximation of subgraph counts in the random-connection model

TL;DR

This paper establishes normal approximation results for subgraph counts in a Poisson-based random-connection model, providing convergence rates and covering various graph regimes.

Contribution

It introduces a combinatorial approach to express cumulants of subgraph counts and derives convergence rates in the Kolmogorov distance for different graph regimes.

Findings

Cumulants expressed via connected partition diagrams.

Convergence rates established using the Statulevičius condition.

Results applicable to dilute, full, and sparse regimes.

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 M\"obius inversion. Using the Statulevi\v{c}ius 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.