Maximal Steiner Trees in the Stochastic Mean-Field Model of Distance

TL;DR

This paper investigates the asymptotic behavior of Steiner trees in a complete graph with exponential edge weights, revealing how their weights scale with the number of vertices for both typical and worst-case scenarios.

Contribution

It extends previous results by characterizing the maximum Steiner tree weight over all vertex sets, including mixed cases of typical and worst-case vertices.

Findings

Worst-case k-Steiner tree weight scales as (2k-1) log n / n.

Typical pairwise distance scales as log n / n.

Results generalize to mixed sets of vertices.

Abstract

Consider the complete graph on $n$ vertices, with edge weights drawn independently from the exponential distribution with unit mean. Janson showed that the typical distance between two vertices scales as $\log{n}/n$, whereas the diameter (maximum distance between any two vertices) scales as $3\log{n}/n$. Bollob\'{a}s et al. showed that, for any fixed k, the weight of the Steiner tree connecting $k$ typical vertices scales as $(k-1)\log{n}/n$, which recovers Janson's result for $k=2$. We extend this result to show that the worst case $k$-Steiner tree, over all choices of $k$ vertices, has weight scaling as $(2k-1)\log{n}/n$ and finally, we generalise this result to Steiner trees with a mixture of typical and worst case vertices.