Open Access
August 2015 Degree distribution of shortest path trees and bias of network sampling algorithms
Shankar Bhamidi, Jesse Goodman, Remco van der Hofstad, Júlia Komjáthy
Ann. Appl. Probab. 25(4): 1780-1826 (August 2015). DOI: 10.1214/14-AAP1036


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. Comput. 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.


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Shankar Bhamidi. Jesse Goodman. Remco van der Hofstad. Júlia Komjáthy. "Degree distribution of shortest path trees and bias of network sampling algorithms." Ann. Appl. Probab. 25 (4) 1780 - 1826, August 2015.


Received: 1 October 2013; Revised: 1 April 2014; Published: August 2015
First available in Project Euclid: 21 May 2015

zbMATH: 1320.60025
MathSciNet: MR3348995
Digital Object Identifier: 10.1214/14-AAP1036

Primary: 05C80 , 60C05 , 90B15

Keywords: Bellman–Harris processes , bias , first passage percolation , Flows , hopcount , mean-field model of distance , network algorithms , power law , random graph , random network , stable-age distribution , weak disorder

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.25 • No. 4 • August 2015
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