  We numerically study the Kuramoto model's synchronization consisting of the
two groups of conformist-contrarian and excitatory-inhibitory phase oscillators
with equal intrinsic frequency. We consider random and small-world (SW)
topologies for the connectivity network of the oscillators. In random networks,
regardless of the contrarian to conformist connection strength ratio, we found
a crossover from the $\pi$-state to the blurred $\pi$-state and then a
continuous transition to the incoherent state by increasing the fraction of
contrarians. However, for the excitatory-inhibitory model in a random network,
we found that for all the values of the fraction of inhibitors, the two groups
remain in-phase, and the transition point of fully synchronized to incoherent
state reduced by strengthening the ratio of inhibitory to excitatory links. In
the SW networks, we found that the order parameters for both models have not
monotonic behavior in terms of the fraction of contrarians and inhibitors. Up
to the optimal fraction of contrarians inhibitors, the synchronization rises by
introducing the contrarians and inhibitors and then falls. We discuss that this
non-monotonic behavior in synchronization is due to the weakening of the
defects formed already in conformists and excitatory agents in SW networks.
