Scaling Laws for Disturbance Propagation in Cyclic Dynamical Networks

TL;DR

This paper investigates how disturbances propagate in cyclic linear networks, providing bounds on their performance measure and showing quadratic scaling with network size for certain classes.

Contribution

It introduces spectral bounds for the $ ext{H}_2$-norm in linear networks and demonstrates quadratic scaling in cyclic network performance as size increases.

Findings

Performance measure bounded by spectral functions of matrices

Quadratic scaling of performance with network size in cyclic networks

Spectral bounds applicable to stable linear dynamical networks

Abstract

Our goal is to analyze performance of stable linear dynamical networks subject to external stochastic disturbances. The square of the $\mathcal H_2$-norm of the network is used as a performance measure to quantify the expected steady-state dispersion of the outputs of the network. We show that this performance measure can be tightly bounded from below and above by some spectral functions of the state-space matrices of the network. This result is applied to a class of cyclic linear networks and shown that their performance measure scale quadratically with the network size.