Connectivity of sparse Bluetooth networks

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 {\sl 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 $n^{-(1-\delta)/d}$ for some $\delta > 0$, then a constant value of $c$ is sufficient for the graph to be connected, with high probability. It suffices to take $c \ge \sqrt{(1+\epsilon)/\delta} + K$ for any positive $\epsilon$ where $K$ is a constant depending on $d$ only. On the other hand, with $c\le \sqrt{(1-\epsilon)/\delta}$, the graph is disconnected, with high probability.