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Multivariate Poisson approximation of joint subgraph counts in random graphs via size-biased couplings

Eulalia Nualart, Rui-Ray Zhang

TL;DR

This work develops a multivariate Poisson approximation framework for joint subgraph counts in Erdős–Rényi graphs using size-biased couplings within the Chen–Stein method, and provides explicit Wasserstein distance bounds. The key novelty is a multivariate size-biased coupling construction in the style of Pianoforte and Turin, yielding bounds that incorporate covariances among subgraph counts. The paper applies the theory to (i) joint counts of fixed strictly balanced subgraphs in $\mathcal{G}(n,p)$ with sharp rates under $p= c n^{-1/\alpha}(1+o(1))$, and (ii) the multivariate hypergeometric distribution, with practical error bounds. It also analyzes the regime $p \to 0$ and discusses extensions to other random graph models and pattern occurrences. Overall, the results provide quantitative, high-dimensional Poisson approximation tools for pattern counting in random graphs, enabling sharper probabilistic descriptions of joint subgraph occurrences.

Abstract

Using Chen-Stein method in combination with size-biased couplings, we obtain the multivariate Poisson approximation in terms of the Wasserstein distance. As applications, we study the multivariate Poisson approximation of the distribution of joint subgraph counts in an Erdős-Rényi random graph and the multivariate hypergeometric distribution giving explicit convergence rates.

Multivariate Poisson approximation of joint subgraph counts in random graphs via size-biased couplings

TL;DR

This work develops a multivariate Poisson approximation framework for joint subgraph counts in Erdős–Rényi graphs using size-biased couplings within the Chen–Stein method, and provides explicit Wasserstein distance bounds. The key novelty is a multivariate size-biased coupling construction in the style of Pianoforte and Turin, yielding bounds that incorporate covariances among subgraph counts. The paper applies the theory to (i) joint counts of fixed strictly balanced subgraphs in with sharp rates under , and (ii) the multivariate hypergeometric distribution, with practical error bounds. It also analyzes the regime and discusses extensions to other random graph models and pattern occurrences. Overall, the results provide quantitative, high-dimensional Poisson approximation tools for pattern counting in random graphs, enabling sharper probabilistic descriptions of joint subgraph occurrences.

Abstract

Using Chen-Stein method in combination with size-biased couplings, we obtain the multivariate Poisson approximation in terms of the Wasserstein distance. As applications, we study the multivariate Poisson approximation of the distribution of joint subgraph counts in an Erdős-Rényi random graph and the multivariate hypergeometric distribution giving explicit convergence rates.
Paper Structure (9 sections, 11 theorems, 56 equations)

This paper contains 9 sections, 11 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Size-biased coupling and Poisson approximation
  3. Multivariate size-biased coupling
  4. Multivariate Poisson approximation
  5. Applications
  6. Joint subgraph counts in random graphs
  7. Proofs of Theorem \ref{['t4']} and Corollary \ref{['t5']}
  8. Strictly balanced graphs
  9. Multivariate hypergeometric distribution

Key Result

Theorem 2.2

R Let $W\geqslant 0$ be an integer-valued random variable with $\mathbb{E}(W)=\lambda>0$, and let $\widetilde{W}$ be a size-biased coupling of $W$. Then, if $P_{\lambda}$ is a Poisson distributed random variable with parameter $\lambda$, we have where the total variation (TV) distance between two distributions $P$ and $Q$ on some finite domain $\mathcal{D}$ is defined by

Theorems & Definitions (28)

  • Definition 2.1: Size-biased coupling
  • Theorem 2.2
  • Definition 2.3: Multivariate size-biased coupling
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 18 more