Minimum Spanning Trees of Random Geometric Graphs with Location Dependent Weights

TL;DR

This paper studies the properties of minimum spanning trees in random geometric graphs with location-dependent edge weights, providing bounds and convergence results for their total weight in the connectivity regime.

Contribution

It introduces a model with location-dependent weights on edges of random geometric graphs and derives deviation bounds and convergence results for the MST total weight.

Findings

Established upper and lower deviation bounds for MST weight.

Proved L^2-convergence of scaled and centered MST weight.

Analyzed behavior in the connectivity regime of the graph.

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.