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Asymptotic normality for triangle counting in the sparse $β$-model

Siang Zhang, Qunqiang Feng, Zhishui Hu

Abstract

We study the number of triangles $T_n$ in the sparse $β$-model on $n$ vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of $T_n$. Next, by applying the Malliavin-Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between normalized $T_n$ and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for $T_n$, as $n\to\infty$.

Asymptotic normality for triangle counting in the sparse $β$-model

Abstract

We study the number of triangles in the sparse -model on vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of . Next, by applying the Malliavin-Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between normalized and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for , as .
Paper Structure (4 sections, 8 theorems, 99 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model description and main results
  3. Mean and variance
  4. Proofs

Key Result

Theorem 1

Under the sparsity condition eq:condion1, we have where $C>0$ is a constant, and

Figures (1)

  • Figure 1: Edge $e_c$ shares a common vertex with two other edges $e_a$ and $e_b$.

Theorems & Definitions (17)

  • Theorem 1
  • Example 1
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 7 more