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arXiv 2005.10922nlin.AO
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网络动力学降维的三重方法：在同步中的应用

Threefold way to the dimension reduction of dynamics on networks: an application to synchronization

Vincent Thibeault, Guillaume St-Onge, Louis J. Dubé, Patrick Desrosiers

备注：
28 pages, 14 figures
发表信息：
Phys. Rev. Research 2, 043215 (2020)
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摘要：

多个复杂系统可以被建模为大型网络，其中节点的状态通过相邻节点之间的相互作用而不断演化，形成高维非线性动力系统。网络科学的主要挑战之一在于预测网络拓扑和动力学对状态演化的影响，特别是对集体现象（如同步）的出现的影响。我们解决这个问题，提出了一个动态近似简化技术（DART），映射高维（完整）的动态到低维（减少）的动态，同时保留最显着的功能，拓扑和动态，原系统。DART通过允许处理复值动态变量、节点内在属性的异质性以及具有强交互社区的模块化网络来概括最近的降维方法。最重要的是，我们确定了三个主要的减少程序，其相对精度取决于是否主要由内在动力学，度序列，或邻接矩阵的状态的演变。我们使用振荡器网络的相位同步作为我们的三重方法的基准。我们成功地预测了随机块模型上的三相动力学（Winfree，Kuramoto，theta）的同步曲线。此外，我们还得到了Kuramoto-Sakaguchi模型在具有非对称块的平均随机块模型上的分支，并数值地证明了在二星图上的外围嵌合态的存在性。这使我们能够突出社区大小的不对称性对嵌合体状态的存在所起的关键作用。最后，我们系统地恢复著名的爆炸同步的分析结果，通过使用DART的Kuramoto-Sakaguchi模型上的星图。

英文摘要：

Several complex systems can be modeled as large networks in which the state of the nodes continuously evolves through interactions among neighboring nodes, forming a high-dimensional nonlinear dynamical system. One of the main challenges of Network Science consists in predicting the impact of network topology and dynamics on the evolution of the states and, especially, on the emergence of collective phenomena, such as synchronization. We address this problem by proposing a Dynamics Approximate Reduction Technique (DART) that maps high-dimensional (complete) dynamics unto low-dimensional (reduced) dynamics while preserving the most salient features, both topological and dynamical, of the original system. DART generalizes recent approaches for dimension reduction by allowing the treatment of complex-valued dynamical variables, heterogeneities in the intrinsic properties of the nodes as well as modular networks with strongly interacting communities. Most importantly, we identify three major reduction procedures whose relative accuracy depends on whether the evolution of the states is mainly determined by the intrinsic dynamics, the degree sequence, or the adjacency matrix. We use phase synchronization of oscillator networks as a benchmark for our threefold method. We successfully predict the synchronization curves for three phase dynamics (Winfree, Kuramoto, theta) on the stochastic block model. Moreover, we obtain the bifurcations of the Kuramoto-Sakaguchi model on the mean stochastic block model with asymmetric blocks and we show numerically the existence of periphery chimera state on the two-star graph. This allows us to highlight the critical role played by the asymmetry of community sizes on the existence of chimera states. Finally, we systematically recover well-known analytical results on explosive synchronization by using DART for the Kuramoto-Sakaguchi model on the star graph.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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