Randomized consensus algorithms over large scale networks

Various randomized consensus algorithms have been proposed in the literature. In some case randomness is due to the choice of a randomized network communication protocol. In other cases, randomness is simply caused by the potential unpredictability of the environment in which the distributed consensus algorithm is implemented. Conditions ensuring the convergence of these algorithms have already been proposed in the literature. As far as the rate of convergence of such algorithms, two approaches can be proposed. One is based on a mean square analysis, while a second is based on the concept of Lyapunov exponent. In this paper, by some concentration results, we prove that the mean square convergence analysis is the right approach when the number of agents is large. Differently from the existing literature, in this paper we do not stick to average preserving algorithms. Instead, we allow to reach consensus at a point which may differ from the average of the initial states. The advantage of such algorithms is that they do not require bidirectional communication among agents and thus they apply to more general contexts. Moreover, in many important contexts it is possible to prove that the displacement from the initial average tends to zero, when the number of agents goes to infinity.