论文标题
随机图上振荡器的全局同步定理
A global synchronization theorem for oscillators on a random graph
论文作者
论文摘要
考虑$ n $相同的库拉莫托振荡器在随机图上。具体来说，考虑\ er随机图，其中任何两个振荡器在双向上与单位强度相结合，独立和随机，概率$ 0 \ leq p \ leq 1 $。我们说，如果振荡器在几乎所有初始条件下都收敛到全相同步状态，则网络正在全球同步。 $ p $以高于哪个全球同步可能极有可能但低于它极为罕见的关键阈值？怀疑存在一个关键阈值，并且接近所谓的连接阈值，即$ p \ sim \ log（n）/n $ for $ n \ gg 1 $。 Ling，Xu和Bandeira取得了第一个进展，以证明结果朝着这个方向证明：他们表明，如果$ p \ gg \ log（n）/n^{1/3} $，那么库拉莫托振荡器的\ er网络在全球范围内以高概率为$ n \ rightarrow \ rightarrow \ rightarrow \ rightarrow \ iffty $。在这里，我们通过证明$ p \ gg \ log^2（n）/n $足够来改善结果。我们的估计是明确的：例如，我们可以说，有$ n = 10^6 $和$ p> 0.01117 $的随机网络有超过$ 99.996 \％$的机会是全球同步的。
Consider $n$ identical Kuramoto oscillators on a random graph. Specifically, consider \ER random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability $0\leq p\leq 1$. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for $p$ above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, $p\sim \log(n)/n$ for $n \gg 1$. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if $p\gg \log(n)/n^{1/3}$, then \ER networks of Kuramoto oscillators are globally synchronizing with high probability as $n\rightarrow\infty$. Here we improve that result by showing that $p\gg \log^2(n)/n$ suffices. Our estimates are explicit: for example, we can say that there is more than a $99.9996\%$ chance that a random network with $n = 10^6$ and $p>0.01117$ is globally synchronizing.
