Extremal Spanning Trees of Random Marked Graphs with Independent Edge Weights

Abstract

In this paper, we consider a Bernoulli random graph~\(G\) on~\(n\) vertices with non-uniform edge probabilities, where each vertex has an independent mark and each edge is equipped with an independent positive weight. The cost of an edge depends on the weight as well the marks of the endvertices and we estimate the growth of the maximum and minimum cost of a spanning tree containing all the vertices. For edge weights with heavy tails and vertex marks distributed uniformly in the unit square, we obtain a phase transition in terms of the tail decay exponent~\(s:\) If~\(s\) is large, then the maximum cost is essentially determined by the vertex locations and if~\(s\) is small, then the edge weights crucially influence the maximum cost. We derive a similar result for minimum cost spanning trees and use martingale difference based methods to establish the~\(L^2-\)convergence of the extremal cost, appropriately scaled and centred.