随机生成图上耦合振子同步预测的学习
Learning to predict synchronization of coupled oscillators on randomly generated graphs
摘要：
假设给定一个耦合振子系统在一个未知图上，以及该系统在某个周期内的轨迹。我们能预测系统最终是否会同步吗？即使有一个已知的底层图结构，这也是一个重要但在分析上难以解决的问题。在这项工作中，我们对同步预测问题采取了另一种方法，将其视为一个分类问题，该问题基于任何给定系统最终将同步或收敛到非同步极限环的事实。通过只使用一些基本的统计数据，如边缘密度和直径，我们的方法可以达到完美的精度时，有一个显着的差异，同步和非同步的例子之间的基础图的拓扑结构。然而，在这些图统计不能很好地区分两个类的问题设置中（例如，当从相同的随机图模型生成图时），我们发现，将初始动态的几次迭代与图统计配对作为我们的分类算法的输入可以导致准确性的显著提高;远远超过经典振荡器理论所知的。更令人惊讶的是，我们发现，在几乎所有这样的设置中，放弃基本的图统计数据，只使用初始动态来训练我们的算法，可以达到几乎相同的精度。我们证明了我们的方法上的三个模型的连续和离散耦合振子-仓本模型，萤火虫细胞自动机，格林伯格-黑斯廷斯模型。最后，我们还提出了一个“集成预测”算法，通过对从多个随机子图观察到的动态进行训练，成功地将我们的方法扩展到大型图。
英文摘要：
Suppose we are given a system of coupled oscillators on an unknown graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? Even with a known underlying graph structure, this is an important yet analytically intractable question in general. In this work, we take an alternative approach to the synchronization prediction problem by viewing it as a classification problem based on the fact that any given system will eventually synchronize or converge to a non-synchronizing limit cycle. By only using some basic statistics of the underlying graphs such as edge density and diameter, our method can achieve perfect accuracy when there is a significant difference in the topology of the underlying graphs between the synchronizing and the non-synchronizing examples. However, in the problem setting where these graph statistics cannot distinguish the two classes very well (e.g., when the graphs are generated from the same random graph model), we find that pairing a few iterations of the initial dynamics along with the graph statistics as the input to our classification algorithms can lead to significant improvement in accuracy; far exceeding what is known by the classical oscillator theory. More surprisingly, we find that in almost all such settings, dropping out the basic graph statistics and training our algorithms with only initial dynamics achieves nearly the same accuracy. We demonstrate our method on three models of continuous and discrete coupled oscillators -- the Kuramoto model, Firefly Cellular Automata, and Greenberg-Hastings model. Finally, we also propose an "ensemble prediction" algorithm that successfully scales our method to large graphs by training on dynamics observed from multiple random subgraphs.
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