The longest minimum-weight path in a complete graph
Louigi Addario-Berry, Nicolas Broutin and Gábor Lugosi
Combinatorics, Probability, and Computing Volume 19, pp. 1-19, 2010.

Abstract

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α⋆ log n edges where α⋆ ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question left open by Janson

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