  The behavior at bifurcation from global synchronization to partial
synchronization in a network of coupled oscillators is a complex phenomenon,
involving the intricate dynamics of one or more oscillators with the bulk of
the remaining synchronized oscillators. This is not captured well by standard
macroscopic model reduction techniques which are designed to capture only the
collective behavior of synchronized oscillators. We introduce two mesoscopic
model reductions for sparse networks of coupled oscillators to quantitatively
capture the dynamics close to bifurcation from global synchronization to
partial synchronization. Our model reduction builds upon the method of
collective coordinates. We first show that standard collective coordinates has
difficulties capturing this bifurcation. We identify a particular topological
structure at bifurcation consisting of a main synchronized cluster, the
oscillator that breaks off at bifurcation, and an intermediary node that
connects them. Utilizing this structure and performing ensemble averages we
derive an analytic expression for the mismatch between the true bifurcation
from global to partial synchronization and its estimate as calculated via the
collective coordinate approach. This allows to calibrate the standard
collective coordinate approach without prior knowledge of which node will break
off. We introduce a second mesoscopic reduction, again utilizing the particular
topological structure of a network close to bifurcation, which allows for a
quantitative dynamical description of the phases near bifurcation, improving
significantly upon standard collective coordinate approaches. We perform
numerical simulations for Erd\H{o}s-R\'enyi networks and for modified
Barab\'asi-Albert networks demonstrating excellent quantitative agreement at
and close to bifurcation.
