  We propose a novel and systematic method for coarse-graining oscillator
networks described by phase equations. Our coarse-graining method enables us to
obtain the closed coarse-grained equations for a few effective eigenmodes,
which is based on the eigenvalue problem of the linearized system around the
phase-locked solution and a nonlinear transformation. We demonstrate our method
by applying it to oscillator dynamics on a random graph, which exhibits a
saddle-node bifurcation at the bifurcation point.
