The longest minimum-weight path in a complete graph

TL;DR

This paper investigates the length of the longest minimum-weight path between node pairs in a complete graph with random exponential edge weights, revealing it scales with approximately 3.59 times log n edges.

Contribution

It provides a precise asymptotic characterization of the longest minimum-weight path length, answering a question posed by Janson (1999).

Findings

Longest minimum-weight path has about 3.5911 log n edges.

The value 3.5911 is the unique solution to alpha log(alpha) - alpha = 1.

Results confirm the asymptotic behavior of path lengths in weighted complete graphs.

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 \alpha^* \log n$ edges where \alpha^* ~ 3.5911 is the unique solution of the equation $alpha log(alpha) - \alpha =1. This answers a question posed by Janson (1999).