Conditional central limit theorems for exponential random graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916799046483968 |
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| author | Fang, Xiao Liu, Song-Hao Su, Zhonggen Wang, Xiaolin |
| author_facet | Fang, Xiao Liu, Song-Hao Su, Zhonggen Wang, Xiaolin |
| contents | In this paper, we study the Exponential Random Graph Models (ERGMs) conditioning on the number of edges. In subcritical region of model parameters, we prove a conditional Central Limit Theorem (CLT) with explicit mean and variance for the number of two stars. This generalizes the corresponding result in the literature for the Erdős--Rényi random graph. To prove our main result, we develop a new conditional CLT via exchangeable pairs based on the ideas of Dey and Terlov. Our key technical contributions in the application to ERGMs include establishing a linearity condition for an exchangeable pair involving two star counts, a local CLT for edge counts, as well as new higher-order concentration inequalities. Our approach also works for general subgraph counts, and we give a conjectured form of their conditional CLT. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conditional central limit theorems for exponential random graphs Fang, Xiao Liu, Song-Hao Su, Zhonggen Wang, Xiaolin Probability 60F05, 05C80 In this paper, we study the Exponential Random Graph Models (ERGMs) conditioning on the number of edges. In subcritical region of model parameters, we prove a conditional Central Limit Theorem (CLT) with explicit mean and variance for the number of two stars. This generalizes the corresponding result in the literature for the Erdős--Rényi random graph. To prove our main result, we develop a new conditional CLT via exchangeable pairs based on the ideas of Dey and Terlov. Our key technical contributions in the application to ERGMs include establishing a linearity condition for an exchangeable pair involving two star counts, a local CLT for edge counts, as well as new higher-order concentration inequalities. Our approach also works for general subgraph counts, and we give a conjectured form of their conditional CLT. |
| title | Conditional central limit theorems for exponential random graphs |
| topic | Probability 60F05, 05C80 |
| url | https://arxiv.org/abs/2506.15159 |