Asymptotic normality for triangle counting in the sparse $β$-model
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912936998469632 |
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| author | Zhang, Siang Feng, Qunqiang Hu, Zhishui |
| author_facet | Zhang, Siang Feng, Qunqiang Hu, Zhishui |
| contents | 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 $n\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01395 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic normality for triangle counting in the sparse $β$-model Zhang, Siang Feng, Qunqiang Hu, Zhishui Probability 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 $n\to\infty$. |
| title | Asymptotic normality for triangle counting in the sparse $β$-model |
| topic | Probability |
| url | https://arxiv.org/abs/2603.01395 |