Transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto model

We investigate transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto (SK) model on complex networks both analytically and numerically. We analytically derive self-consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit. Using the self-consistent equations we investigate transition to synchronization in SK model on uncorrelated scale-free (SF) and Erdős-Rényi (ER) networks in detail. Depending on the degree distribution exponent ($\gamma$) of SF networks and phase-frustration parameter, the population undergoes from first-order transition [explosive synchronization (ES)] to second-order transition and vice versa. In ER networks transition is always second order irrespective of the values of the phase-lag parameter. We observe that the critical coupling strength for the onset of synchronization is decreased by phase-frustration parameter in case of SF network where as in ER network, the phase-frustration delays the onset of synchronization. Extensive numerical simulations using SF and ER networks are performed to validate the analytical results. An analytical expression of critical coupling strength for the onset of synchronization is also derived from the self-consistent equations considering the vanishing order parameter limit.

Figure 1

Synchronization diagram of scale-free network of size $N=1000$, $\gamma =2.8$, and $\langle k\rangle =30$ for four values of $\alpha$. The lag parameter $\alpha$ apparently inhibiting explosive synchronization.

Figure 2

Synchronization diagram of ER network of size $N=1000$ and $\langle k\rangle =30$ for three values of $\alpha$. Higher phase-lag($\alpha$) delays the onset of synchronization.

Figure 3

Effect of $\alpha$ on transition to synchronization for a scale-free network of size $N=10000$ with exponent $\gamma =2.7$ computed using Eqs. (17, 18, 19). (a) Phase diagram on $\alpha \text{−}\lambda$ plane shows asynchronous (cyan, $r\sim 0$), partially synchronous (green) and hysteresis (red) regions delimited by the solid back curve marking the critical coupling strength for spontaneous transition to synchrony (asynchrony) during forward (${\lambda }_f$) and backward (${\lambda }_c$) continuation respectively. Note that (${\lambda }_f={\lambda }_c$) when the value of $\alpha$ is greater than a threshold $(\alpha >0.37)$. Blue triangles and yellow dots are calculated through numerical simulation during forward and backward continuation, respectively. Blue triangles and yellow dots are merged to each other beyond the threshold of $\alpha$. (b) Order parameter $r$ as a function of the coupling strength $\lambda$ for two values of $\alpha$. Solid magenta (for $\alpha =0.5$) and green (for $\alpha =0.1$) curves represent analytically calculated order parameters corresponding stable states and dashed green curve represents the metastable state indicating the width of ES. Square marked solid curves are obtained from numerical integration (magenta for $\alpha =0.5$ and blue for $\alpha =0.1$). (c) Graph of $\text{R}\left(x\right)$ as a function of $x$ for $\alpha =0.1$ (solid green curve) and 0.5 (solid magenta curve). The top and bottom dashed green lines correspond to ${\lambda }_c\left(0.1\right)$ and ${\lambda }_f\left(0.1\right)$ indicating the width of the hysteresis (shown by red arrow).

Figure 4

Effect of $\alpha$ on transition to synchronization for a scale-free network of size $N=10000$ with exponent $\gamma =3.2$ computed using Eqs. (17, 18, 19). (a) Phase diagram on $\alpha \text{−}\lambda$ plane shows asynchronous (cyan, $r\sim 0$) and partially synchronous (green, $r>0$) regions delimited by the solid back curve marking the critical coupling strength (${\lambda }_c$) for spontaneous transition to synchrony. Blue triangles represent the critical coupling strength obtained from numerical simulations. (b) Variation of the order parameter $r$ with coupling strength $\lambda$ for two values of $\alpha$. Solid curves (green and magenta) are drawn based on analytical approach and squares (green and magenta) are obtained from numerical simulations.

Figure 5

Effect of $\alpha$ on transition to synchronization for an ER of size $N=10000$ computed using Eqs. (17, 18, 19). (a) Phase diagram on $\alpha \text{−}\lambda$ plane shows asynchronous (cyan, $r\sim 0$) and partially synchronous (green, $r>0$) regions delimited by the solid back curve marking the critical coupling strength (${\lambda }_c$) for spontaneous transition to synchrony. Blue triangles represent the critical coupling strength obtained from numerical simulations. (b) Variation of the order parameter $r$ with coupling strength $\lambda$ for two values of $\alpha$. Solid curves (green and magenta) are drawn based on analytical approach and squares (green and magenta) are obtained from numerical simulations.

Figure 6

Critical coupling strength as a function of $\alpha$ for two SF and one ER networks of size $N=10000$ and mean degree $\langle k\rangle =30$ computed from Eqs. (20) and (21). Solid cyan and red curves are used for SF networks with $\gamma =2.7$ and 3.2, respectively, while solid blue curve is used for ER network.