We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\mathbb{T}^n \rightarrow \mathbb{R}$ be given by $$ f(\theta_1, \dots, \theta_n) = \sum_{i,j=1}^{n}{ a_{ij} \cos{(\theta_i - \theta_j)}}.$$ This function has a global maximum when $\theta_i = \theta$ for all $1\leq i \leq n$. It is known that if every vertex is connected to at least $\mu(n-1)$ other vertices for $\mu$ sufficiently large, then every local maximum is global. Taylor proved this for $\mu \geq 0.9395$ and Ling, Xu \& Bandeira improved this to $\mu \geq 0.7929$. We give a slight improvement to $\mu \geq 0.7889$. Townsend, Stillman \& Strogatz suggested that the critical value might be $\mu_c = 0.75$.

中文翻译：

---

密集网络中仓本振荡器的同步

我们研究仓本振荡器系统的同步特性。该问题也可以理解为关于由图形创建的能源景观的属性的问题。更正式地说，让 $G=(V,E)$ 是一个连通图，$(a_{ij})_{i,j=1}^{n}$ 表示它的邻接矩阵。令函数 $f:\mathbb{T}^n \rightarrow \mathbb{R}$ 由 $$ f(\theta_1, \dots, \theta_n) = \sum_{i,j=1}^{n }{ a_{ij} \cos{(\theta_i - \theta_j)}}.$$ 当$\theta_i = \theta$ 对于所有$1\leq i \leq n$ 时，该函数具有全局最大值。众所周知，如果每个顶点连接到至少 $\mu(n-1)$ 个其他顶点且 $\mu$ 足够大，那么每个局部最大值都是全局的。Taylor 用 $\mu \geq 0.9395$ 证明了这一点，Ling, Xu \& Bandeira 将其改进为 $\mu \geq 0.7929$。我们对 $\mu \geq 0.7889$ 略有改进。汤森，