Distributed Average Consensus in Sensor Networks with Random Link Failures and Communication Channel Noise

Abstract

In this paper we study distributed average consensus type algorithms in sensor networks with random network link failures and communication channel. Specifically, the network links fail randomly across iterations, and communication through an active link incurs additive stochastic noise. We consider the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ND</i> algorithm for distributed average consensus under such imperfect communication scenario. Using results from the theory of controlled Markov processes and stochastic approximation, we show that the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ND</i> algorithm leads to consensus of the sensor states. In particular, all the sensor states converge a.s. to a finite random variable thetas, the latter being an unbiased estimate of the desired average. We explicitly characterize the resulting the mean-squared error (m.s.e.) and show that the m.s.e. can be made arbitrarily small by tuning certain parameters of the algorithm. But, reducing the m.s.e. in this way, decrease the convergence rate of the algorithm, and we obtain an interesting trade-off between the m.s.e. and the convergence rate of the algorithm. Our results show that the sensor network topology plays a significant role in determining the convergence rate of these algorithms.