Monotone properties of random geometric graphs have sharp thresholds

• 成果类型: Article
• 署名作者: Goel, A; Rai, S; Krishnamachari, B
• 署名单位: Stanford University; Stanford University; Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051605000000575
• 发表日期: 2005
• 页码: 2535-2552
• 关键词: connectivity

摘要

Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0, 1](d), and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching oil two sets of n points distributed uniformly in [0, 1](d). We present upper bounds on the threshold width, and show that our bound is sharp for d = 1 and at most a sublogarithmic factor away for d >= 2. Interestingly, the threshold. width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.