A Frequency Domain Analysis of Slow Coherency in Networked Systems

TL;DR

This paper introduces a frequency domain framework to analyze network coherence in linear systems, linking it to low-rank properties of the transfer matrix and revealing how increasing connectivity fosters coherent behavior.

Contribution

It develops a novel frequency domain approach to quantify network coherence, relating it to the rank of the transfer matrix and providing insights into the role of network topology and dynamics.

Findings

Coherence increases with effective algebraic connectivity.

The transfer matrix converges to a rank-one matrix representing coherence.

Harmonic mean of nodal dynamics determines the coherent behavior.

Abstract

Network coherence generally refers to the emergence of simple aggregated dynamical behaviours, despite heterogeneity in the dynamics of the subsystems that constitute the network. In this paper, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system's input-output response. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network's \emph{effective algebraic connectivity} grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as the coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.