2014/02/15 by Broutin, Nicolas, Devroye, Luc, Lugosi, Gábor
#05C80 #60C05 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Networking and Internet Architecture (cs.NI) #Probability (math.PR)
paper · doi:10.48550/arxiv.1402.3696
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" rn. 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-δ)/d for some δ> 0, then a constant value of c is sufficient for the graph to be connected, with high probability. It suffices to take c ≥ √((1+ε)/δ) + K for any positive ε where K is a constant depending on d only. On the other hand, with c≤ √((1-ε)/δ), the graph is disconnected, with high probability.