One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights

Abstract

Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n/n for two given points, that the maximum if one point is fixed and the other varies is 2 log n/n, and that the maximum over all pairs of points is 3 log n/n.Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths.