Mean Squared Error Analysis of Noisy Average Consensus

A continuous-time average consensus system is a linear dynamical system defined over a graph, where each node has its own state value that evolves according to a simultaneous linear differential equation. A node is allowed to interact with neighboring nodes. Average consensus is a phenomenon that the all the state values converge to the average of the initial state values. In this paper, we assume that a node can communicate with neighboring nodes through an additive white Gaussian noise channel. We first formulate the noisy average consensus system by using a stochastic differential equation (SDE), which allows us to use the Euler-Maruyama method, a numerical technique for solving SDEs. By studying the stochastic behavior of the residual error of the Euler-Maruyama method, we arrive at the covariance evolution equation. The analysis of the residual error leads to a compact formula for mean squared error (MSE), which shows that the sum of the inverse eigenvalues of the Laplacian matrix is the most dominant factor influencing the MSE.