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 language | English (US) |
|---|---|
| Pages (from-to) | 656-672 |
| Number of pages | 17 |
| Journal | SIAM Journal on Discrete Mathematics |
| Volume | 28 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2014 |
| Externally published | Yes |
Fingerprint
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 journal › Article
}
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 -