The Min Mean-Weight Cycle in a Random Network

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 meanweight 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 Θp(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/π2 − o(1)) log n log log n.