Minimum Spanning Trees of Random Geometric Graphs with Location Dependent Weights
Abstract
Consider~\(n\) nodes~\(\Xi\1 ≤ i ≤ n\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(Xi\) and~\(Xj\) are joined by an edge if the Euclidean distance~\(d(Xi,Xj)\) is less than~\(rn,\) the adjacency distance and the resulting random graph~\(Gn\) is called a random geometric graph~(RGG). We now assign a location dependent weight to each edge of~\(Gn\) and define~\(MSTn\) to be the sum of the weights of the minimum spanning trees of all components of~\(Gn.\) For values of~\(rn\) above the connectivity regime, we obtain upper and lower bound deviation estimates for~\(MSTn\) and~\(L2-\)convergence of~\(MSTn\) appropriately scaled and centred.
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