Shortest-Weight paths in random regular graphs

Leo Hamed Amini, Yuval Peres

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.

Original languageEnglish (US)
Pages (from-to)656-672
Number of pages17
JournalSIAM Journal on Discrete Mathematics
Volume28
Issue number2
DOIs
StatePublished - 2014
Externally publishedYes

Fingerprint

Regular Graph
Random Graphs
Precise Asymptotics
Path
Identically distributed
Unique Solution
Assign
Random variable
Vertex of a graph

Keywords

  • First passage percolation
  • Shortest-weight paths
  • Weighted random graphs

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Shortest-Weight paths in random regular graphs. / Amini, Leo Hamed; Peres, Yuval.

In: SIAM Journal on Discrete Mathematics, Vol. 28, No. 2, 2014, p. 656-672.

Research output: Contribution to journalArticle

@article{17b13a5ddd264c43b670515c876a4c4b,
title = "Shortest-Weight paths in random regular graphs",
abstract = "Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.",
keywords = "First passage percolation, Shortest-weight paths, Weighted random graphs",
author = "Amini, {Leo Hamed} and Yuval Peres",
year = "2014",
doi = "10.1137/120899534",
language = "English (US)",
volume = "28",
pages = "656--672",
journal = "SIAM Journal on Discrete Mathematics",
issn = "0895-4801",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - Shortest-Weight paths in random regular graphs

AU - Amini, Leo Hamed

AU - Peres, Yuval

PY - 2014

Y1 - 2014

N2 - Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.

AB - Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.

KW - First passage percolation

KW - Shortest-weight paths

KW - Weighted random graphs

UR - http://www.scopus.com/inward/record.url?scp=84904018647&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904018647&partnerID=8YFLogxK

U2 - 10.1137/120899534

DO - 10.1137/120899534

M3 - Article

AN - SCOPUS:84904018647

VL - 28

SP - 656

EP - 672

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -