A large-deviations principle for all the components in a sparse inhomogeneous random graph

• 成果类型: Article
• 署名作者: Andreis, Luisa; Koenig, Wolfgang; Langhammer, Heide; Patterson, Robert I. A.
• 署名单位: Polytechnic University of Milan; Technical University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01180-7
• 发表日期: 2023
• 页码: 521-620
• 关键词: scaling limits; coagulation; inversion; models

摘要

We study an inhomogeneous sparse random graph, G(N), on [N] = {1, . . . , N} as introduced in a seminal paper by Bollobas et al. (Random Struct Algorithms 31(1):3-122, 2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N -> infinity, we consider the sparse regime, where the average degree is O (1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size asymptotic to N). In doing so, we derive explicit logarithmic asymptotics for the probability that G(N) is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of G(N). In particular, we recover the criterion for the existence of the phase transition given in Bollobas et al. (2007).