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On the Critical Value for 'Percolation' of Minimum-Weight Trees in the Mean-Field Distance Model

Abstract

Consider the complete n-graph with independent exponential (mean n) edge-weights. Let M(c, n) be the maximal size of subtree for which the average edge-weight is at most c. It is shown that M(c, n) makes the transition from o(n) to Ω(n) around some critical value c(0), which can be specified in terms of a fixed point of a mapping on probability distributions.

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