Euclidean minimum spanning trees with location dependent and power weighted edges
Abstract
Consider~\(n\) nodes~\(\Xi\1 ≤ i ≤ n\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f\) and let~\(Kn\) be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge~\(e\) in~\(Kn\) we associate a weight~\(w(e)\) that may depend on the individual locations of the endvertices of~\(e.\) Denoting~\(MSTn\) to be the minimum weight of a spanning tree of~\(Kn\) and assuming an equivalence condition on the weight function~\(w(.),\) we prove that~\(MSTn\) appropriately scaled and centred converges to zero a.s.\ and in mean as~\(n → ∞.\) We also obtain upper and lower bound deviation estimates for~\(MSTn.\)
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