  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.
