  We have compared the phase synchronization transition of the second order
Kuramoto model on 2D lattices and on large, synthetic power grid networks,
generated from real data. The latter are weighted, hierarchical modular
networks. Due to the inertia the synchronization transitions are of first order
type, characterized by fast relaxation and hysteresis by varying the global
coupling parameter K. Finite size scaling analysis shows that there is no real
phase transition in the thermodynamic limit, unlike in the mean-field model.
The order parameter and its fluctuations depend on the network size without any
real singular behavior. In case of power grids the phase synchronization breaks
down at lower global couplings, than in case of 2D lattices of the same sizes,
but the hysteresis is much narrower or negligible due to the low connectivity
of the graphs. The temporal behavior of de-synchronization avalanches after a
sudden quench to low K values, has been followed and duration distributions
with power-law tails have been detected. This suggests rare region effects,
caused by frozen disorder, resulting in heavy tailed distributions, even
without a self organization mechanism as a consequence of a catastrophic drop
event in the couplings.
