Probabilistic properties of highly connected random geometric graphs

Abstract

We study the probabilistic properties of reliable networks of minimum costs in d-dimensional Euclidean space, with reliability in terms of k-edge-connectivity in graphs. We show that this problem fits into Yukich's framework for Euclidean functionals for arbitrary k, dimension d and distant-power gradient p with p < d. With this framework, several theorems on convergence and concentration of the value of optimal solutions follow.These results are then extended to optimal k-edge-connected power assignment graphs, where we assign transmit power to nodes, and two nodes are connected if they both have sufficient transmit power. This variant models wireless networks.Finally, we devise a partitioning heuristic to find approximate solutions quickly, and we analyze its performance in the framework of smoothed analysis.