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Network coherence generally refers to the emergence of simple aggregateddynamical behaviours, despite heterogeneity in the dynamics of the subsystemsthat constitute the network. In this paper, we develop a general frequencydomain framework to analyze and quantify the level of network coherence that asystem exhibits by relating coherence with a low-rank property of the system'sinput-output response. More precisely, for a networked system with lineardynamics and coupling, we show that, as the network's \emph{effective algebraicconnectivity} grows, the system transfer matrix converges to a rank-onetransfer matrix representing the coherent behavior. Interestingly, the non-zeroeigenvalue of such a rank-one matrix is given by the harmonic mean ofindividual nodal dynamics, and we refer to it as the coherent dynamics. Ouranalysis unveils the frequency-dependent nature of coherence and a non-trivialinterplay between dynamics and network topology. We further show that manynetworked systems can exhibit similar coherent behavior by establishing aconcentration result in a setting with randomly chosen individual nodaldynamics.

A Frequency Domain Analysis of Slow Coherency in Networked Systems