  This paper focuses on the consensus averaging problem on graphs under general
noisy channels. 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 exact consensus for finite
variance noise. Our analysis applies to various types of stochastic
disturbances, including errors in parameters, transmission noise, and
quantization noise. Under a suitable stability condition, we prove that the
error is asymptotically Gaussian, and we show how the asymptotic covariance is
specified by the graph Laplacian. For additive parameter noise, we show how the
scaling of the asymptotic MSE is controlled by the spectral gap of the
Laplacian.
