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The Min Mean-Weight Cycle in a Random Network

Published online by Cambridge University Press:  22 July 2013

CLAIRE MATHIEU  and
DAVID B. WILSON
CLAIRE MATHIEU
Affiliation:
Brown University and Ecole Normale Supérieure, DI ENS, CNRS UMR 8548, 45 rue d'Ulm, 75005 Paris, France (e-mail: clairemmathieu@gmail.com)
DAVID B. WILSON
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: david.wilson@microsoft.com)
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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 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)) log2n log log n.

Keywords

Primary 05C80Secondary 68Q87

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 5 , September 2013 , pp. 763 - 782
Copyright
Copyright © Cambridge University Press 2013 

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