On the performance of broadcast based consensus under non-zero-mean stochastic disturbances

The broadcast-based consensus algorithm is one special type of randomized consensus algorithm, and is amenable to practical implementation in wireless networks. This paper focuses on the performance analysis of this broadcast-based consensus algorithm in the presence of non-zero-mean stochastic perturbations. It is demonstrated that as the algorithm proceeds, the deviation of the node states from their average will converge, in expectation, to a fixed value, which is determined by the Laplacian matrix of the network, the mixing parameter, and the mean of the stochastic perturbations. Asymptotic upper and lower bounds on the total mean-square deviation are derived, which describe the range of distances over which the node states deviate from consensus. These bounds can facilitate evaluation of the applicability of this algorithm in practice. In addition, performance of the broadcast-based consensus algorithm under zero-mean stochastic disturbances is also analyzed, and results regarding its convergence and mean-square deviation are given. The theoretical results presented in this work hold true regardless of the statistics of the stochastic disturbances, and are valid for arbitrary network topology as long as the topology is connected.