Minimum Spanning Trees of Random Geometric Graphs with Location Dependent Weights (2103.00764v1)

Abstract: 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.