The min mean-weight cycle in a random network

TL;DR

This paper analyzes the distribution and properties of the minimum mean-weight cycle in a complete random graph with exponential edge weights, revealing phase transitions and cycle length behaviors as the graph size grows.

Contribution

It provides a detailed probabilistic analysis of the minimum mean-weight cycle in a random complete graph, identifying phase transitions and cycle length distributions.

Findings

Probability of min mean weight ≤ c/n converges to a limiting function.

Cycle length distribution changes at the threshold c=1/e.

For weights > 1/(en), cycles are typically long, at least proportional to log^2 n.

Abstract

The mean weight of a cycle in an edge-weighted graph is the sum of the cycle's edge weights divided by the cycle's length. We study the minimum mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge weights drawn from an exponential distribution with mean 1. We show that the probability of the min mean weight being at most c/n tends to a limiting function of c which is analytic for c<=1/e, discontinuous at c=1/e, and equal to 1 for c>1/e. We further show that if the min mean weight is <=1/(en), then the length of the relevant cycle is Theta_p(1) (i.e., it has a limiting probability distribution which does not scale with n), but that if the min mean weight is >1/(en), then the relevant cycle almost always has mean weight (1+o(1))/(en) and length at least (2/pi^2-o(1)) log^2 n log log n.