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Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

Mindaugas Bloznelis, Joona Karjalainen, Lasse Leskelä

TL;DR

This work analyzes counts of dense subgraphs in a clustered, sparse random network formed as a superposition of Bernoulli layers with random sizes and strengths. The authors develop normal and $\alpha$-stable limit theorems for copies of $2$-connected patterns, with precise moment conditions on the layer size $X$ and strength $Q$, and without requiring $X$ and $Q$ to be independent. The normal limit arises when the per-layer counts have finite variance, while heavy-tailed per-layer contributions induce a stable law with index $\alpha\in(0,2)$ after centering; the results cover general $2$-connected graphs and special cases like cliques. The proofs combine a layered-decomposition approach, tight control of overlaps, and classical limit theorems (CLT and Gnedenko–Kolmogorov stable limits), providing a rigorous link between clustering, power-law degrees, and subgraph count distributions in complex networks.

Abstract

The clustering property of complex networks indicates the abundance of small dense subgraphs in otherwise sparse networks. For a community-affiliation network defined by a superposition of Bernoulli random graphs, which has a nonvanishing global clustering coefficient and a power-law degree distribution, we establish normal and $α$--stable approximations to the number of small cliques, cycles and more general $2$-connected subgraphs.

Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

TL;DR

This work analyzes counts of dense subgraphs in a clustered, sparse random network formed as a superposition of Bernoulli layers with random sizes and strengths. The authors develop normal and -stable limit theorems for copies of -connected patterns, with precise moment conditions on the layer size and strength , and without requiring and to be independent. The normal limit arises when the per-layer counts have finite variance, while heavy-tailed per-layer contributions induce a stable law with index after centering; the results cover general -connected graphs and special cases like cliques. The proofs combine a layered-decomposition approach, tight control of overlaps, and classical limit theorems (CLT and Gnedenko–Kolmogorov stable limits), providing a rigorous link between clustering, power-law degrees, and subgraph count distributions in complex networks.

Abstract

The clustering property of complex networks indicates the abundance of small dense subgraphs in otherwise sparse networks. For a community-affiliation network defined by a superposition of Bernoulli random graphs, which has a nonvanishing global clustering coefficient and a power-law degree distribution, we establish normal and --stable approximations to the number of small cliques, cycles and more general -connected subgraphs.

Paper Structure

This paper contains 8 sections, 8 theorems, 78 equations.

Table of Contents

  1. Introduction and results
  2. Network model
  3. Results
  4. Proofs
  5. Notation
  6. Proofs
  7. Auxiliary lemmas
  8. Power-law tails

Key Result

Theorem 1

Let $\nu>0$. Let $n,m\to+\infty$ and assume that $m/n\to\nu$. Let $F$ be a $2$-connected graph with $v_F\ge 3$ vertices. Assume that ${\bf{E}} X<\infty$ and $0< \sigma^2_{F}<\infty$. Assume, in addition, that Then $({\cal N}_F-{\bf{E}} {\cal N}_F)/(\sigma_{F}\sqrt{m})$ converges in distribution to the standard normal distribution.

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Corollary 1
  • proof : Proof of Theorem \ref{['normal_limit_B']} and Remark \ref{['remark-clique_normal']}
  • proof : Proof of Theorem \ref{['stable_limit_B']} and Remark \ref{['remark-clique_stable']}
  • proof : Proof of Remark \ref{['remark-1']}
  • Lemma 1
  • ...and 9 more