  We study the synchronized state in a population of network-coupled,
heterogeneous oscillators. In particular, we show that the steady-state
solution of the linearized dynamics may be written as a geometric series whose
subsequent terms represent different spatial scales of the network. Namely,
each addition term incorporates contributions from wider network neighborhoods.
We prove that this geometric expansion converges for arbitrary frequency
distributions and for both undirected and directed networks provided that the
adjacency matrix is primitive. We also show that the error in the truncated
series grows geometrically with the second largest eigenvalue of the normalized
adjacency matrix, analogously to the rate of convergence to the stationary
distribution of a random walk. Lastly, we derive a local approximation for the
synchronized state by truncating the spatial series, at the first neighborhood
term, to illustrate the practical advantages of our approach.
