Distributed Consensus Algorithms in Sensor Networks: Link Failures and Channel Noise

We study average consensus when, simultaneously, the topology is random (links are offline or online at random times) and the communication among sensors is corrupted by additive noise. Additive noise causes the states of the standard average consensus algorithm to diverge. To overcome this, we consider two modifications to average consensus: \begin{inparaenum}[1)] \item the $\mathcal{A-ND}$ algorithm with weights decaying to zero (slowly, satisfying a persistence condition); and \item the $\mathcal{A-NC}$ algorithm with time invariant weights but that averages successive runs, restarted with the same initial conditions. \end{inparaenum} To study the behavior of these two algorithms under the simultaneous random link failures and additive noise, we use controlled Markov processes and stochastic approximation results. With respect to the $\mathcal{A-ND}$ algorithm, we show that the states reach a.s. consensus to a finite random variable, whose variance can be made arbitrarily small, and the expected value of the states reach consensus to the desired average of the initial states. Regarding the $\mathcal{A-NC}$ algorithm, we prove that it reaches $\epsilon$-consensus, i.e., for sufficiently large number of sufficiently long runs, the sensor states reach within an $\epsilon$-ball the desired average with high probability. The paper characterizes analytically, and illustrates through numerical simulations, the tradeoffs among the network and the two algorithms parameters: signal-to-noise ratio, algebraic connectivity, weight sequence, rate of convergence to consensus, variance of the consensus random variable, number of iterations, and number of runs.