Mean Square Limitations of Spatially Invariant Networked Systems

In this chapter, we summarize some of our recent results on spatially invariant networked systems. We study the effects of unreliable communication on these systems' stability and performance. Because of their special structure, the quantities that characterize the limitations of these systems can be computed more easily. In particular, we focus on Mean Square stability and performance, and investigate network architectures, which are more robust and better performing. We consider multi-agent networked systems where the communication links are unreliable and stochastically dropout. Spatial invariance leads to a simplified computation of the MS stability limitation, and allows us to derive an uncertainty conservation law enjoyed by such systems. We then focus on distributed averaging systems, for which the loss of Mean Square stability leads to the emergence of certain complex behavior related to Lévy flights. We present closed form formulae characterizing Mean Square stability and performance in the presence of unreliable communication among the nodes. Finally, we study Mean square performance in the presence of unreliable links. Our results allow to characterize the interplay between Mean Square performance and stability of torus networks of different dimensions.