  We study the synchronization of a small-world network of identical coupled
phase oscillators with Kuramoto interaction. First, we consider the model with
instantaneous mutual interaction and the normalized coupling constant to the
degree of each node. For this model, similar to the constant coupling studied
before, we find the existence of various attractors corresponding to the
different defect patterns and also the noise enhanced synchronization when
driven by an external uncorrelated white noise. We also investigate the
synchronization of the model with homogenous time-delay in the phase couplings.
For a given intrinsic frequency and coupling constant, upon varying the time
delay we observe the existence a partially synchronized state with defect
patterns which transforms to an incoherent phase characterized by randomly
phase locked states. By further increasing of the time delay, this phase again
undergoes a transition to another patterned partially synchronized state. We
show that the transition between theses phases are discontinuous and moreover
in each phase the average frequency of the oscillators decreases by increasing
the time delay and shows an upward jump at the transition points.
