Connectivity of sparse bluetooth networks

Abstract

Consider a random geometric graph defined on n vertices uniformly distributedin the d-dimensional unit torus. Two vertices are connected if their distance is less than a "visibility radius" rn. 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)/dfor some δ>0, then a constant value of c is sufficient forthe graph to be connected, with high probability. It suffices to takec≥(1+ϵ)/δ−−−−−−−√+K for any positive ϵ where Kis a constant depending on d only. On the other hand, with c≤(1−ϵ)/δ−−−−−−−√,the graph is disconnected, with high probability.