On the mean square error of randomized averaging algorithms

This paper regards randomized discrete-time consensus systems that preserve the average on expectation. As a main result, we provide an upper bound on the mean square deviation of the consensus value from the initial average. Then, we particularize our result to systems where the interactions which take place simultaneously are few, or weakly correlated; these assumptions cover several algorithms proposed in the literature. For such systems we show that, when the system size grows, the deviation tends to zero, not slower than the inverse of the size. Our results are based on a new approach, unrelated to the convergence properties of the system: this independence questions the relevance in this context of the spectral properties of the matrices related to the graph of possible interactions, which have appeared in some previous results.