Euclidean minimum spanning trees with location dependent and power weighted edges

Consider~$n$ nodes~$\{X_i\}_{1 \leq i \leq n}$ independently distributed in the unit square~$S,$ each according to a distribution~$f$ and let~$K_n$ be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge~$e$ in~$K_n$ we associate a weight~$w(e)$ that may depend on the \emph{individual locations} of the endvertices of~$e.$ Denoting~$MST_n$ to be the minimum weight of a spanning tree of~$K_n$ and assuming an equivalence condition on the weight function~$w(.),$ we prove that~$MST_n$ appropriately scaled and centred converges to zero a.s.\ and in mean as~$n \rightarrow \infty.$ We also obtain upper and lower bound deviation estimates for~$MST_n.$