TYPICAL STRUCTURE OF SPARSE EXPONENTIAL RANDOM GRAPH MODELS
成果类型：
Article
署名作者：
Cook, Nicholas A.; Dembo, Amir
署名单位：
Duke University; Stanford University
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/23-AAP2025
发表日期：
2024
页码：
2885-2939
关键词：
摘要：
We consider general exponential random graph models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs F-k. Whereas previous work has shown a degeneracy phenomenon in dense ERGMs, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the naive mean-field approximation for the partition function of the corresponding Gibbs measures, and in case of ferromagnetic models with vanishing edge density show that typical samples resemble a typical Erdos-Renyi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erdos-Renyi graph in the large deviations regime for excess F-k-homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of (Adv. Math. 319 (2017) 313-347) on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner's generalized Holder inequality.
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