PERCOLATION ON DENSE GRAPH SEQUENCES

• 成果类型: Article
• 署名作者: Bollobas, Bela; Borgs, Christian; Chayes, Jennifer; Riordan, Oliver
• 署名单位: University of Cambridge; University of Memphis; Microsoft; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP478
• 发表日期: 2010
• 页码: 150-183
• 关键词: random subgraphs; finite graphs; PHASE-TRANSITION; component; EVOLUTION

摘要

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G(n)). Let lambda(n) be the largest eigenvalue of the adjacency matrix of G(n), and let G(n)(p(n)) be the random subgraph of G(n) obtained by keeping each edge independently with probability p(n). We show that the appearance of a giant component in G(n)(P-n) has a sharp threshold at p(n) = 1/lambda(n). In fact, we prove much more: if (G(n)) converges to an irreducible limit, then the density of the largest component of G(n)(c/n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovasz, Sos and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollobas, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.