随机图上振子的全局同步定理
A global synchronization theorem for oscillators on a random graph
摘要：
考虑随机图上的$n$全同仓本振子。具体地说，考虑\ER随机图，其中任何两个振荡器都是双向耦合的单位强度，独立和随机，概率$0\leq p\leq 1$。我们说一个网络是全局同步的，如果振荡器收敛到几乎所有的初始条件的全同相同步状态。$p$是否存在一个临界阈值，高于该阈值，全局同步极有可能发生，但低于该阈值，全局同步极罕见？怀疑存在临界阈值并且接近所谓的连通性阈值，即，对于$n \gg 1$的$p\sim \log(n)/n$。Ling、Xu和Bandeira在证明这个方向的结果方面取得了第一个进展：他们证明，如果$p\gg \log(n)/n^{1/3}$，那么仓本振子的\ER网络以高概率全局同步为$n\rightarrow\infty$。在这里，我们通过证明$p\gg \log^2(n)/n$是足够的来改进该结果。我们的估计是明确的：例如，我们可以说，具有$n = 10^6$和$p>0.01117$的随机网络全局同步的可能性大于$99.9996\%$。
英文摘要：
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.
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