  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.
