  We consider the Kuramoto model on sparse random networks such as the
Erd\H{o}s-R\'enyi graph or its combination with a regular two-dimensional
lattice and study the dynamical scaling behavior of the model at the
synchronization transition by large-scale, massively parallel numerical
integration. By this method, we obtain an estimate of critical coupling
strength more accurate than obtained earlier by finite-size scaling of the
stationary order parameter. Our results confirm the compatibility of the
correlation-size and the temporal correlation-length exponent with the
mean-field universality class. However, the scaling of the order parameter
exhibits corrections much stronger than those of the Kuramoto model with
all-to-all coupling, making thereby an accurate estimate of the order-parameter
exponent hard. We find furthermore that, as a qualitative difference to the
model with all-to-all coupling, the effective critical exponents involving the
order-parameter exponent, such as the effective decay exponent characterizing
the critical desynchronization dynamics show a non-monotonic approach toward
the asymptotic value. In the light of these results, the technique of
finite-size scaling of limited size data for the Kuramoto model on sparse
graphs has to be treated cautiously.
