Contrariety and inhibition enhance synchronization in a small-world network of phase oscillators

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.