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) |
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Pages (from-to) | 656-672 |
Number of pages | 17 |
Journal | SIAM Journal on Discrete Mathematics |
Volume | 28 |
Issue number | 2 |
DOIs | |
State | Published - 2014 |
Keywords
- First passage percolation
- Shortest-weight paths
- Weighted random graphs
ASJC Scopus subject areas
- Mathematics(all)