On the Spatial Pattern of Input-Output Metrics for a Network Synchronization Process

TL;DR

This paper provides a graph-theoretic analysis of input-output metrics in network synchronization, revealing spatial degradation patterns and introducing a concept of propagation stability in noisy diffusive networks.

Contribution

It introduces a novel spatial analysis of transfer metrics in synchronization models and characterizes signal propagation and stability in noisy networks.

Findings

Transfer metrics decrease along vertex cutsets from inputs.

Spatial degradation pattern is formally characterized.

Propagation stability concept is developed for noisy networks.

Abstract

A graph-theoretic analysis is undertaken for a compendium of input-output (transfer) metrics of a standard discrete-time linear synchronization model, including lp gains, frequency responses, frequency-band energy, and Markov parameters. We show that these transfer metrics exhibit a spatial degradation, such that they are monotonically nonincreasing along vertex cutsets away from an exogenous input. We use this spatial analysis to characterize signal-to-noise ratios (SNRs) in diffusive networks driven by process noise, and to develop a notion of propagation stability for dynamical networks. Finally, the formal results are illustrated through an example.