Open Access
2015 Connectivity of sparse bluetooth networks
Nicolas Broutin, Luc Devroye, Gabor Lugosi
Author Affiliations +
Electron. Commun. Probab. 20: 1-10 (2015). DOI: 10.1214/ECP.v20-3644


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" $r_n$. 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 $n^{-(1-\delta)/d}$for some $\delta > 0$, then a constant value of $c$ is sufficient forthe 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.


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Nicolas Broutin. Luc Devroye. Gabor Lugosi. "Connectivity of sparse bluetooth networks." Electron. Commun. Probab. 20 1 - 10, 2015.


Accepted: 12 June 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1319.05114
MathSciNet: MR3358970
Digital Object Identifier: 10.1214/ECP.v20-3644

Primary: 05C80
Secondary: 60C05

Keywords: connectivity , irrigation graph , Random geometric graph

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