Connectivity of sparse bluetooth networks

Broutin, Nicolas; Devroye, Luc; Lugosi, Gábor

doi:http://dx.doi.org/10.1214/ECP.v20-3644

Connectivity of sparse bluetooth networks

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Broutin N, Devroye L, Lugosi G. Connectivity of sparse bluetooth networks. Electron Commun Probab. 2015;20(48):1-10. DOI: 10.1214/ECP.v20-3644

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.