Department of Mathematics, National Institute of Technology, Durgapur 713209, India.
Chaos. 2019 Jan;29(1):013123. doi: 10.1063/1.5045836.
We investigate transition to synchronization in the Sakaguchi-Kuramoto (SK) model on complex networks analytically as well as numerically. Natural frequencies of a percentage (f) of higher degree nodes of the network are assumed to be correlated with their degrees and that of the remaining nodes are drawn from some standard distribution, namely, Lorentz distribution. The effects of variation of f and phase frustration parameter α on transition to synchronization are investigated in detail. Self-consistent equations involving critical coupling strength (λ) and group angular velocity (Ω) at the onset of synchronization have been derived analytically in the thermodynamic limit. For the detailed investigation, we considered the SK model on scale-free (SF) as well as Erdős-Rényi (ER) networks. Interestingly, explosive synchronization (ES) has been observed in both networks for different ranges of values of α and f. For SF networks, as the value of f is set within 10%≤f≤70%, the range of the values of α for existence of the ES is greatly enhanced compared to the fully degree-frequency correlated case when scaling exponent γ<3. ES is also observed in SF networks with γ>3, which is never observed in fully degree-frequency correlated environment. On the other hand, for random networks, ES observed is in a narrow window of α when the value of f is taken within 30%≤f≤50%. In all the cases, critical coupling strengths for transition to synchronization computed from the analytically derived self-consistent equations show a very good agreement with the numerical results. Finally, we observe ES in the metabolic network of the roundworm Caenorhabditis elegans in partially degree-frequency correlated environment.
我们从理论和数值上研究了 Sakaguchi-Kuramoto (SK) 模型在复杂网络中的同步转变。假设网络中一定比例（f）的高度数节点的自然频率与其度数相关，而其余节点的自然频率则来自于某种标准分布，即洛伦兹分布。我们详细研究了 f 和相位受挫参数α的变化对同步转变的影响。在热力学极限下，我们从理论上推导出了自洽方程，其中包含了同步开始时的临界耦合强度（λ）和群体角速度（Ω）。为了进行详细的研究，我们考虑了在无标度（SF）和 Erdos-Renyi（ER）网络上的 SK 模型。有趣的是，对于不同的α和 f 值范围，我们在这两种网络中都观察到了爆炸同步（ES）。对于 SF 网络，当 f 值设定在 10%≤f≤70%时，与完全度频相关情况相比，存在 ES 的α值范围大大增强，而当标度指数γ<3 时。在γ>3 的 SF 网络中也观察到了 ES，而在完全度频相关环境中从未观察到 ES。另一方面，对于随机网络，当 f 值在 30%≤f≤50%之间时，ES 出现在一个狭窄的α窗口中。在所有情况下，从理论上推导出来的自洽方程计算得到的同步转变的临界耦合强度与数值结果非常吻合。最后，我们在秀丽隐杆线虫 Caenorhabditis elegans 的代谢网络中观察到了部分度频相关环境中的 ES。
