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Low degree connectivity of ad-hoc networks via percolation

Part of:Equilibrium statistical mechanicsLimit theoremsComputer system organization

Published online by Cambridge University Press:  01 July 2016

Emilio De Santis,
Fabrizio Grandoni and
Alessandro Panconesi
Emilio De Santis*
Affiliation:
Sapienza University of Rome
Fabrizio Grandoni*
Affiliation:
Sapienza University of Rome
Alessandro Panconesi*
Affiliation:
Sapienza University of Rome
*
∗ Postal address: Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy. Email address: desantis@mat.uniroma1.it
∗∗ Current address: DISP, Tor Vergata University of Rome, Via del Politecnico 1, 00191 Rome, Italy. Email address: grandoni@disp.uniroma2.it
∗∗∗ Postal address: Department of Computer Science, Sapienza University of Rome, Via Salaria 113, 00198 Rome, Italy. Email address: ale@di.uniroma1.it
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Abstract

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Consider the following classical problem in ad-hoc networks. Suppose that n devices are distributed uniformly at random in a given region. Each device is allowed to choose its own transmission radius, and two devices can communicate if and only if they are within the transmission radius of each other. The aim is to (quickly) establish a connected network of low average and maximum degree. In this paper we present the first efficient distributed protocols that, in poly-logarithmically many rounds and with high probability, set up a connected network with O(1) average degree and O(log n) maximum degree. Our algorithms are based on the following result, which is a nontrivial consequence of classical percolation theory. Suppose that each device sets up its transmission radius in order to reach the K closest devices. There exists a universal constant K (independent of n) such that, with high probability, there will be a unique giant component (i.e. a connected component of size Θ(n)). Furthermore, all remaining components will be of size O(log2 n). This leads to an efficient distributed probabilistic test for membership in the giant component, which can be used in a second phase to achieve full connectivity.

Keywords

Networksconnectivitypercolation

MSC classification

Primary:68M10: Network design and communication82B43: Percolation
Secondary:60F10: Large deviations

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010, pp. 559 - 576
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

A preliminary version of this work was published in Algorithms - ESA 2007 (Proc. 15th Annual Europ. Symp. Algorithms, 2007; Lecture Notes Comput. Sci. 4698), Springer, Berlin, pp. 206-217.

This work was done when the author was at the Department of Computer Science of Sapienza University.

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