On the Stretch Factor of Randomly Embedded Random Graphs

TL;DR

This paper analyzes the stretch factor of randomly embedded Erdős–Rényi graphs, providing bounds and conditions for bounded stretch factor as the number of vertices grows.

Contribution

It establishes asymptotic bounds on the stretch factor of embedded random graphs and characterizes when it remains bounded, answering a previously open question.

Findings

Stretch factor is bounded with high probability if and only if n(1-p) tends to 0.

Provides asymptotically tight upper and lower bounds on the stretch factor.

Shows bounds are optimal for certain ranges of p.

Abstract

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all u,v in V. We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for p not too close to 0 or 1, these bounds are best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if n(1-p) tends to 0, answering a question of O'Rourke.