论文标题
扩展器图是全球同步的
Expander graphs are globally synchronizing
论文作者
论文摘要
Kuramoto模型是同步研究的基础。它由一个振荡器集合组成，这些振荡器具有由网络给出的相互作用的集合，我们分别使用图形的顶点和边缘识别。在本文中，我们表明，具有足够膨胀的图必须是全球同步的，这意味着该图上相同振荡器的均匀的库拉莫托模型将收敛到完全同步的状态，所有具有相同阶段的振荡器，对于每个初始状态的每个初始状态均可达到零阶段。特别是，我们表明，对于任何$ \ varepsilon> 0 $和$ p \ geq（1 + \ varepsilon）（\ log n） / n $，erdős-rényi随机图$ g（n，p）上的同质kuramoto模型全球范围内与概率同步，符合一个$ n $ n $ n $ n $ n nim $ n os n os n os n os n os n os $ n。这改善了Kassabov，Strogatz和Townsend的先前结果，并解决了Ling，Xu和Bandeira的猜想。我们还表明，该模型在任何$ d $的Ramanujan图上都在全球同步，并且在典型的$ d $ -Regrumar-graphs上，对于足够大的$ d $。
The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p \geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the Erdős-Rényi random graph $G(n, p)$ is globally synchronizing with probability tending to one as $n$ goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the model is globally synchronizing on any $d$-regular Ramanujan graph, and on typical $d$-regular graphs, for large enough degree $d$.
