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Normal approximation for number of edges in random intersection graphs

Katarzyna Rybarczyk, Grzegorz Serafin

TL;DR

This work analyzes normal approximation for the edge count in the random intersection graph ${\\mathcal G}(n,m,p)$. It develops a general bound for local statistics via a Stein–Malliavin–type approach applied to chaos decompositions, and introduces clique-cover based combinatorics to bound central moments of subgraph indicators. The authors apply the method to the standardized edge count $\\widetilde{N}_E$, deriving necessary and sufficient conditions for asymptotic normality and providing sharp convergence rates in the sparse regime $mp^3\le 1$ as well as partial results for larger $p$. A flexible framework linking contraction norms to subgraph probabilities through clique covers enables analysis beyond single edges and yields threshold-type behavior in parameter regimes, with potential extensions to larger cliques. Overall, the paper unifies probabilistic and combinatorial techniques to address dependence in random intersection graphs and offers practical bounds across a wide range of parameters.

Abstract

The random intersection graph model $\mathcal G(n,m,p)$ is considered. Due to substantial edge dependencies, studying even fundamental statistics such as the subgraph count is significantly more challenging than in the classical binomial model $\mathcal G(n,p)$. First, we establish normal approximation bound in both the Wasserstein and the Kolmogorov distances for a class of local statistics on $\mathcal G(n,m,p)$. Next, we apply these results to derive such bounds for the standardised number of edges, and determine the necessary and sufficient conditions for its asymptotic normality. We develop a new method that provides a combinatorial interpretation and facilitates the estimation of analytical expressions related to general distance bounds. In particular, this allows us to control the behaviour of central moments of subgraph existence indicators. The presented method can also be extended to count copies of subgraphs larger than a single edge.

Normal approximation for number of edges in random intersection graphs

TL;DR

This work analyzes normal approximation for the edge count in the random intersection graph . It develops a general bound for local statistics via a Stein–Malliavin–type approach applied to chaos decompositions, and introduces clique-cover based combinatorics to bound central moments of subgraph indicators. The authors apply the method to the standardized edge count , deriving necessary and sufficient conditions for asymptotic normality and providing sharp convergence rates in the sparse regime as well as partial results for larger . A flexible framework linking contraction norms to subgraph probabilities through clique covers enables analysis beyond single edges and yields threshold-type behavior in parameter regimes, with potential extensions to larger cliques. Overall, the paper unifies probabilistic and combinatorial techniques to address dependence in random intersection graphs and offers practical bounds across a wide range of parameters.

Abstract

The random intersection graph model is considered. Due to substantial edge dependencies, studying even fundamental statistics such as the subgraph count is significantly more challenging than in the classical binomial model . First, we establish normal approximation bound in both the Wasserstein and the Kolmogorov distances for a class of local statistics on . Next, we apply these results to derive such bounds for the standardised number of edges, and determine the necessary and sufficient conditions for its asymptotic normality. We develop a new method that provides a combinatorial interpretation and facilitates the estimation of analytical expressions related to general distance bounds. In particular, this allows us to control the behaviour of central moments of subgraph existence indicators. The presented method can also be extended to count copies of subgraphs larger than a single edge.

Paper Structure

This paper contains 9 sections, 18 theorems, 200 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results and discussion
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:prob->discrete']}
  5. Clique covers and subgraph probabilities
  6. Graphic representations of integrals
  7. Asymptotic normality for edge count
  8. Distance bounds
  9. Asymptotic normality conditions

Key Result

Theorem 2.1

Let $X$ be of the form eq:statistic with symmetric $h\in L^4((\{0, 1\}^m)^r, \mu_{m, p}^{\otimes r})$. Then we have where $C_r$ is a constant depending on $r$, and

Figures (5)

  • Figure 1: Graphs $G_i$, $i=1, 2, 3$.
  • Figure 2: Examples of $H_{1,{\bf I}}$. $G_{i,{\bf I}}$ has been placed in the dotted rectangle.
  • Figure 3: Examples of $H_{1,{\bf I}, {\bf J}}$, where ${\bf I}=\{11, 12, 21\}$ and ${\bf J}={\bf\Gamma}, \{11, 12, 21\}, \{21, 22\}$.
  • Figure 4: Let ${\bf J}=\{11, 12, 21\}$ and ${\bf J}^*=\{11,21\}$. In the figure we show all $H_{1,{\bf I}}$ such that ${\bf J}\cap{\bf I}={\bf J}^*$. By \ref{['Eq:HIIpromHIIbis']} all $H_{1,{\bf I}, {\bf J}}$ (with the edge set indicated by the dotted line) are isomorphic. Each $H_{1,{\bf I}, {\bf J}}$ consist of $G_{{\bf J}^*}$ (built with edges $e_{11}$ and $e_{21}$), an isolated edge $e_{12}$, and three isolated vertices (those outside the dotted line).
  • Figure 5: Examples of clique covers without 2--sets for $H_{i,{\bf\Gamma}}$.

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • proof : Proof of Theorem \ref{['thm:prob->discrete']}
  • Lemma 5.1
  • ...and 26 more