## The Annals of Applied Probability

### Degree distribution of shortest path trees and bias of network sampling algorithms

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 4 (2015), 1780-1826.

Dates
Revised: April 2014
First available in Project Euclid: 21 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1432212430

Digital Object Identifier
doi:10.1214/14-AAP1036

Mathematical Reviews number (MathSciNet)
MR3348995

Zentralblatt MATH identifier
1320.60025

#### Citation

Bhamidi, Shankar; Goodman, Jesse; van der Hofstad, Remco; Komjáthy, Júlia. Degree distribution of shortest path trees and bias of network sampling algorithms. Ann. Appl. Probab. 25 (2015), no. 4, 1780--1826. doi:10.1214/14-AAP1036. https://projecteuclid.org/euclid.aoap/1432212430

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