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This paper focuses on the consensus averaging problem on graphs under general imperfect communications. We study a particular class of distributed consensus algorithms based on damped updates, and using the ordinary differential equation method, we prove that the updates converge almost surely to the consensus average for various models of perturbation of data exchanged between nodes. The convergence is not asymptotic in the size of the network. Our analysis applies to various types of stochastic disturbances to the updates, including errors in update calculations, dithered quantization and imperfect data exchange among nodes. Under a suitable stability condition, we prove that the error between estimated and true averages is asymptotically Gaussian, and we show how the asymptotic covariance is specified by the graph Laplacian. For additive perturbations, we show how the scaling of the asymptotic MSE is controlled by the spectral gap of the Laplacian.
Date of Publication: Jan. 2011