  Network topology plays an important role in governing the collective
dynamics. Partial synchronization (PaS) on regular networks with a few
non-local links is explored. Different PaS patterns out of the symmetry
breaking are observed for different ways of non-local couplings. The criterion
for the emergence of PaS is studied. The emergence of PaS is related to the
loss of degeneration in Lyapunov exponent spectrum. Theoretical and numerical
analysis indicate that non-local coupling may drastically change the dynamical
feature of the network, emphasizing the important topological dependence of
collective dynamics on complex networks.
