DEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS

• 成果类型: Article
• 署名作者: Bhamidi, Shankar; Goodman, Jesse; van der Hofstad, Remco; Komjathy, Julia
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Technion Israel Institute of Technology; Eindhoven University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1036
• 发表日期: 2015
• 页码: 1780-1826
• 关键词: 1st passage percolation; Random graphs; asymptotics; TOPOLOGY; point

摘要

In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large degrees of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance), as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random r-regular graphs for large r, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean-field model of distance. We use our results to shed light on an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [Ann. Appl. Probab. 20 (2010) 1907 - 1965], [Combin. Probab. Cornput. 20 (2011) 683-707], [Adv. in AppL Probab. 42 (2010) 706-738] of analyzing the effect of attaching random edge lengths on the geometry of random network models.