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 onlineat random times) and the communication among sensors is corrupted by additive noise. Additive noisecauses the states of the standard average consensus algorithm to diverge. To overcome this, we considertwo modifications to average consensus: 1) the A−ND algorithm with weights decaying to zero (slowly,satisfying a persistence condition); and 2) the A−NC algorithm with time invariant weights but thataverages successive runs, restarted with the same initial conditions. To study the behavior of these twoalgorithms under the simultaneous random link failures and additive noise, we use controlled Markovprocesses and stochastic approximation results.With respect to the A−ND algorithm, we show that the states reach a.s. consensus to a finiterandom variable, whose variance can be made arbitrarily small, and the expected value of the statesreach consensus to the desired average of the initial states. Regarding the A−NC algorithm, we provethat it reaches ǫ-consensus, i.e., for sufficiently large number of sufficiently long runs, the sensor statesreach within an ǫ-ball the desired average with high probability. The paper characterizes analytically, andillustrates through numerical simulations, the tradeoffs among the network and the two algorithms pa-rameters: 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.