Coherence and Concentration in Tightly-Connected Networks

TL;DR

This paper introduces a framework linking network coherence to low-rank properties of the system, showing that increased connectivity leads to coherent behavior characterized by a rank-one transfer matrix, with frequency-dependent effects.

Contribution

It develops a general approach to analyze and quantify network coherence, relating it to the spectral properties of the system and demonstrating convergence to a rank-one transfer matrix as connectivity increases.

Findings

Network coherence relates to a low-rank property of the system.

As connectivity grows, the transfer matrix converges to a rank-one matrix.

Frequency influences the degree of coherence in the network.

Abstract

The ability to achieve coordinated behavior -- engineered or emergent -- on networked systems has attracted widespread interest over several fields. This interest has led to remarkable advances in developing a theoretical understanding of the conditions under which agents within a network can reach an agreement (consensus) or develop coordinated behavior, such as synchronization. However, much less understood is the phenomenon of network coherence. Network coherence generally refers to nodes' ability in a network to have a similar dynamic response despite heterogeneity in their individual behavior. In this paper, we develop a general 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. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network 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.