Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917301630009344 |
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| author | Chatterjee, Sayak Chatterjee, Anirban Chakraborty, Abhinav Bhattacharya, Bhaswar B. |
| author_facet | Chatterjee, Sayak Chatterjee, Anirban Chakraborty, Abhinav Bhattacharya, Bhaswar B. |
| contents | In this paper, we derive the asymptotic distribution of the number of copies of a fixed graph $H$ in a random graph $G_n$ sampled from a sparse graphon model. Specifically, we provide a refined analysis that separates the contributions of edge randomness and vertex-label randomness, allowing us to identify distinct sparsity regimes in which each component dominates or both contribute jointly to the fluctuations. As a result, we establish asymptotic normality for the count of any fixed graph $H$ in $G_n$ across the entire range of sparsity (above the containment threshold for $H$ in $G_n$). These results provide a complete description of subgraph count fluctuations in sparse inhomogeneous networks, closing several gaps in the existing literature that were limited to specific motifs or suboptimal sparsity assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs Chatterjee, Sayak Chatterjee, Anirban Chakraborty, Abhinav Bhattacharya, Bhaswar B. Probability Statistics Theory In this paper, we derive the asymptotic distribution of the number of copies of a fixed graph $H$ in a random graph $G_n$ sampled from a sparse graphon model. Specifically, we provide a refined analysis that separates the contributions of edge randomness and vertex-label randomness, allowing us to identify distinct sparsity regimes in which each component dominates or both contribute jointly to the fluctuations. As a result, we establish asymptotic normality for the count of any fixed graph $H$ in $G_n$ across the entire range of sparsity (above the containment threshold for $H$ in $G_n$). These results provide a complete description of subgraph count fluctuations in sparse inhomogeneous networks, closing several gaps in the existing literature that were limited to specific motifs or suboptimal sparsity assumptions. |
| title | Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2512.12937 |