Broadcast-Based Consensus With Non-Zero-Mean Stochastic Perturbations

Bolstered by the growing interest in building wireless sensor and ad hoc networks with applications ranging across different engineering disciplines, distributed consensus algorithms have recently seen a new revival since their inception in the early 1980s. Of particular interest is the recently developed broadcast-based consensus algorithm, which is one special type of randomized consensus algorithms 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. Results are also provided on the algorithm's ε-converging time, i.e., the earliest time at which the deviation is ε close to its steady value, and on the mean and mean-square behaviors of the displacement of node states from their initial states at large iteration number. As a special case study, performance of the broadcast-based consensus algorithm under zero-mean stochastic disturbances is analyzed, and results regarding its convergence, mean-square deviation, and mean-square displacement are given. The theoretical results presented in this study hold true regardless of the statistics of the stochastic disturbances, and are valid for arbitrary network topology as long as the topology is connected.