On finding a minimum spanning tree in a network with random weights

We investigate Prim’s standard ‘tree-growing’ method for finding a minimum spanning tree, when applied to a network in which all degrees are about d and the edges e have independent identically distributed random weights w(e). We find that when the kth edge ek is added to the current tree, where k = o( √ d), the probability that this edge ek is incident to the node that was most recently added to the tree equals 12 + 1 2k + o(1) as d → ∞. We also find for example that, if the edge weights are uniformly distributed on (0, 1), then the expected value of w(ek) is asymptotic to (12 + 1 2k )/d.