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

We numerically study Kuramoto model 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 an 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 do not show monotonic behavior in terms of the fraction of contrarians and inhibitors. Up to the optimal fraction of contrarians and inhibitors, the synchronization rises by introducing the number of contrarians and inhibitors and then falls. We discuss that the nonmonotonic behavior in synchronization is due to the weakening of the defects already formed in the pure conformist and excitatory agent model in SW networks. We found that in SW networks, the optimal fraction of contrarians and inhibitors remain unchanged for the rewiring probabilities up to $\sim 0.15$, above which synchronization falls monotonically, like the random network. We also showed that in the conformist-contrarian model, the optimal fraction of contrarians is independent of the strength of contrarian links. However, in the excitatory-inhibitory model, the optimal fraction of inhibitors is approximately proportional to the inverse of inhibition strength.

Figure 1

Stationary total and partial order parameters vs fraction of contrarians for the conformist-contrarian model [Eq. (2)] for (a) $Q=0.5$, (b) $Q=1.0$, and (c) $Q=3.0$ for a random network of $N=1000$ oscillators and mean degree $<k>=10$.

Figure 2

Stationary total and partial order parameters vs fraction of inhibitory oscillators for the excitatory-inhibitory [Eq. (4)] model for (a) $Q=0.5$, (b) $Q=1.0$, and (c) $Q=3.0$ for a random network of $N=1000$ oscillators and mean degree $<k>=10$.

Figure 3

Stationary total and partial order parameters vs fraction of contrarians for the conformist-contrarian model [Eq. (2)] for (a) $Q=0.5$, (b) $Q=1.0$, and (c) $Q=3.0$ for a SW network of $N=1000$ oscillators and mean degree $<k>=10$.

Figure 4

Stationary total and partial order parameters vs fraction of inhibitory oscillators for the excitatory-inhibitory [Eq. (4)] model for (a) $Q=0.5$, (b) $Q=1.0$, and (c) $Q=3.0$ for a SW network of $N=1000$ oscillators and mean degree $<k>=10$.

Figure 5

Temporal evolution of the phases of rotors in a single realization of the conformist-contrarian model [Eq. (2)], with $Q=1.0$, in an SW network for (a) $p=0.0$, (b) $p=0.03$, (c) $p=0.05$, and (d) $p=0.09$.

Figure 6

Temporal evolution of the phases of rotors in a single realization of the excitatory-inhibitory model [Eq. (4)], with $Q=1.0$, in an SW network for (a) $p=0.0$, (b) $p=0.03$, (c) $p=0.05$, and (d) $p=0.09$.

Figure 7

Log-log plot of optimal fraction of inhibitory oscillators ($p_m$) vs their link strength $Q$ in a small-world network of excitatory-inhibitory rotors of size $N=1000$ and mean degree $<k>=10$ and with rewiring probabilities $p=0.03$.

Figure 8

Stationary order parameters vs fraction of contrarians for the conformist-contrarian model [Eq. (2)] for $Q=1$ and in a small-world of $N=1000$ oscillators and mean degree $<k>=10$ with rewiring probabilities $p_r=0.03,0.05,0.09,0.15$.