论文标题
保证在随机几何图上自发同步
Guarantees for Spontaneous Synchronization on Random Geometric Graphs
论文作者
论文摘要
Kuramoto模型是非线性动力学系统领域中的经典数学模型，它描述了可能达到同步状态的网络中耦合振荡器的演变。网络的拓扑与振荡器同步之间的关系是同步领域的主要问题，并且随机图通常被用作复杂网络的代理。另一方面，文献中严格分析库拉莫托模型的随机图是同质模型，并且无法捕获几个示例中出现的基本几何结构。 在这项工作中，我们利用从随机矩阵理论，随机图和数学统计数据中利用工具来证明球体上随机几何图上的Kuramoto模型随着概率倾向于一个淋巴结的数量倾向于无限。据我们所知，这是随机几何图上库拉莫托模型的第一个严格结果。
The Kuramoto model is a classical mathematical model in the field of non-linear dynamical systems that describes the evolution of coupled oscillators in a network that may reach a synchronous state. The relationship between the network's topology and whether the oscillators synchronize is a central question in the field of synchronization, and random graphs are often employed as a proxy for complex networks. On the other hand, the random graphs on which the Kuramoto model is rigorously analyzed in the literature are homogeneous models and fail to capture the underlying geometric structure that appears in several examples. In this work, we leverage tools from random matrix theory, random graphs, and mathematical statistics to prove that the Kuramoto model on a random geometric graph on the sphere synchronizes with probability tending to one as the number of nodes tends to infinity. To the best of our knowledge, this is the first rigorous result for the Kuramoto model on random geometric graphs.
