Connectivity of sparse bluetooth networks

Abstract

Consider a random geometric graph defined on n vertices uniformly distributed in the d-dimensional unit torus. Two vertices are connected if their distance is less than a “visibility radius” r n. We consider Bluetooth networks that are locally sparsified random geometric graphs. Each vertex selects c of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of n -(1- δ )/d for some d > 0, then a constant value of c is sufficient for the graph to be connected, with high probability. It suffices to take c (Formula Present)K for any positive Є where K is a constant depending on d only. On the other hand, with c ≤ (Formula Present), the graph is disconnected, with high probability.