Effects of assortative mixing in the second-order Kuramoto model

In this paper we analyze the second-order Kuramoto model in the presence of a positive correlation between the heterogeneity of the connections and the natural frequencies in scale-free networks. We numerically show that discontinuous transitions emerge not just in disassortative but also in strongly assortative networks, in contrast with the first-order model. We also find that the effect of assortativity on network synchronization can be compensated by adjusting the phase damping. Our results show that it is possible to control collective behavior of damped Kuramoto oscillators by tuning the network structure or by adjusting the dissipation related to the phases' movement.

Figure 1

Synchronization diagram $r\left(\lambda \right)$ with (a) $\alpha =0.2$, (b) $\alpha =0.4$, (c) $\alpha =0.6$, and (d) $\alpha =0.8$ for assortativity $\mathcal{A}=-0.3$. With increasing $\alpha$, onset of synchronization and hysteresis decrease. The natural frequency of each oscillator is ${\Omega }_i=k_i-\langle k\rangle$ and the networks have $N={10}^3$ and $\langle k\rangle =6$. The degree distribution follows a power law $P\left(k\right)\sim k^{-\gamma }$, where $\gamma =3$. Curves in which points are connected by solid lines (dashed lines) correspond to the forward (backward) continuations of the coupling strength $\lambda$.

Figure 2

Synchronization diagram $r\left(\lambda \right)$ for (a) $\mathcal{A}=-0.3$, (b) $\mathcal{A}=-0.1$, (c) $\mathcal{A}=0.1$, and (d) $\mathcal{A}=0.3$. The dissipation coefficient is fixed at $\alpha =1$ with natural frequencies given by ${\Omega }_i=k_i-\langle k\rangle$, as in Fig. 1. All networks considered have $N={10}^3,\langle k\rangle =6$, and $P\left(k\right)\sim k^{-\gamma }$, where $\gamma =3$.

Figure 3

Contour plot on $\alpha \text{−}\mathcal{A}$ plane colored according to (a) the maximal order parameter $\mathcal{MO}$, (b) the maximal order parameter difference $\mathcal{MD}$, and (c) the hysteresis area $\mathcal{S}$. $\mathcal{A}$ and $\alpha$ are varied within the interval $[-0.3,0.3]$ and $[0.2,1]$, respectively. For each pair $(\alpha ,\mathcal{A})$, 40 times simulations are performed with the coupling strength in the interval as in Fig. 1, i.e., $\lambda \in [0,25]$.

Figure 4

Parameters are the same as in Fig. 1, except that the natural frequencies are randomly selected from a Lorentzian distribution $g\left(\Omega \right)=1/\left[\pi (1+{\Omega }^2)\right]$.

Figure 5

Parameters are the same as in Fig. 2, except that natural frequencies are randomly selected from a Lorentzian distribution $g\left(\Omega \right)=1/\left[\pi (1+{\Omega }^2)\right]$.

Figure 6

Contour plots similar as in Fig. 3 for networks with a natural frequency distribution given by $g\left(\Omega \right)=1/\left[\pi (1+{\Omega }^2)\right]$.