The min mean-weight cycle in a random network

Details The min mean-weight cycle in a random network The min mean-weight cycle in a random network

2012-01-19

arxiv.org/abs/1201.3955v1

Mathematics

Probability

20 pages, 1 figure

Scientific paper

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.

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