RANDOM NETWORKS WITH SUBLINEAR PREFERENTIAL ATTACHMENT: THE GIANT COMPONENT

• 成果类型: Article
• 署名作者: Dereich, Steffen; Moerters, Peter
• 署名单位: University of Munster; University of Bath
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP697
• 发表日期: 2013
• 页码: 329-384
• 关键词: phase-transition; random graph; CONVERGENCE

摘要

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a milltitype branching random walk associated with f.