Minimum Spanning Trees of Random Geometric Graphs with Location Dependent Weights

Consider~$n$ nodes~$\{X_i\}_{1 \leq i \leq n}$ independently distributed in the unit square~$S,$ each according to a distribution~$f.$ Nodes~$X_i$ and~$X_j$ are joined by an edge if the Euclidean distance~$d(X_i,X_j)$ is less than~$r_n,$ the adjacency distance and the resulting random graph~$G_n$ is called a random geometric graph~(RGG). We now assign a location dependent weight to each edge of~$G_n$ and define~$MST_n$ to be the sum of the weights of the minimum spanning trees of all components of~$G_n.$ For values of~$r_n$ above the connectivity regime, we obtain upper and lower bound deviation estimates for~$MST_n$ and~$L^2-$convergence of~$MST_n$ appropriately scaled and centred.