Edge Flows in the Complete Random-Lengths Network

TL;DR

This paper analyzes the distribution of flows on edges in a complete graph with random exponential edge lengths, revealing the limiting empirical distribution of edge-flows as the number of vertices grows large.

Contribution

It provides an explicit characterization of the asymptotic empirical distribution of edge-flows in the complete random-lengths network.

Findings

Derived the limiting distribution of edge-flows as n approaches infinity.

Established the normalization needed for the empirical distribution.

Provided explicit formulas for the distribution in the large network limit.

Abstract

Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the $n \to \infty$ limit we find explicitly the empirical distribution of these edge-flows, suitably normalized.