Mean Square Performance and Robust Yet Fragile Nature of Torus Networked Average Consensus

This paper addresses the mean square (MS) performance problem for torus networked average consensus under stochastic disturbances. We investigate how a torus networked average consensus will behave when all of the communication links are unreliable and each node is subject to the perturbation of additive noise. We first show that the MS performance problem is equivalent to an MS stability problem, which can be further transformed into a standard norm computation. Although the analysis complexity does not scale nicely with system size and dimension, based on the spatially invariant structure of torus lattices as well as the decomposition of their Laplacians, we are able to derive an efficiently computable closed-form formula for analysis. That is, we express the MS performance index in terms of the eigenvalues of the 1-D torus Laplacian, making the MS performance analysis efficient. We discover, for a fixed update gain, that the torus networked average consensus of a high dimension performs the best when in the MS stability regime, but it is also the most vulnerable to MS instability. This reflects a tradeoff between MS stability and performance.