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Asymptotic normality for triangle counting in the sparse $\beta$-model

Siang Zhang et al.

Journal of Applied Probability2026https://doi.org/10.1017/jpr.2026.10090article
AJG 2ABDC A
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0.50

Abstract

We study the number of triangles $T_n$ in the sparse $\beta$ -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 the 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 $n\to\infty$ .

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https://doi.org/https://doi.org/10.1017/jpr.2026.10090

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@article{siang2026,
  title        = {{Asymptotic normality for triangle counting in the sparse $\beta$-model}},
  author       = {Siang Zhang et al.},
  journal      = {Journal of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1017/jpr.2026.10090},
}

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Asymptotic normality for triangle counting in the sparse $\beta$-model

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