  We propose a degree-based coarse graining approach that not just accelerates
the evaluation of dynamics on complex networks, but also satisfies the
consistency conditions for both equilibrium statistical distributions and
nonequilibrium dynamical flows. For the Ising model and
susceptible-infected-susceptible epidemic model, we introduce these required
conditions explicitly and further prove that they are satisfied by our
coarse-grained network construction within the annealed network approximation.
Finally, we numerically show that the phase transitions and fluctuations on the
coarse-grained network are all in good agreements with those on the original
one.
