Low degree connectivity of ad-hoc networks via percolation

De Santis, Emilio and Grandoni, Fabrizio and Panconesi, Alessandro (2010) Low degree connectivity of ad-hoc networks via percolation. Advances in Applied Probability, 42 (2). pp. 559-576. ISSN 0001-8678

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Consider the following classical problem in ad-hoc networks: 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 non-trivial consequence of classical percolation theory: suppose that all devices set up their 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. aconnected component of size Θ(n)). Furthermore, all remaining components will be of size 1 O(log 2n). 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.

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