Large deviations of empirical neighborhood distribution in sparse random graphs

• 成果类型: Article
• 署名作者: Bordenave, Charles; Caputo, Pietro
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; Roma Tre University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0590-8
• 发表日期: 2015
• 页码: 149-222
• 关键词: convergent sequences; PRINCIPLE

摘要

Consider the ErdAs-Renyi random graph on vertices where each edge is present independently with probability , with fixed. For large , a typical random graph locally behaves like a Galton-Watson tree with Poisson offspring distribution with mean . Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right.