序参量允许基于仓本模型中的平衡不动点对平面图进行分类
Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model
摘要：
相位平衡态是仓本模型和网络上其他耦合振荡器模型的一类高度未开发的解决方案。到目前为止，耦合振荡器的研究主要集中在相位同步的解决方案。然而，对振子的全局约束可能禁止同步状态，从而使相位平衡状态成为相关的稳定状态。如果例如振荡器驱动流体填充体积的收缩，则流体体积的守恒将振荡器约束到平衡状态，如由消失的Kuramoto序参量表征的。以前已经证明，在循环图中存在稳定的，平衡的模式在仓本模型。然而，哪些非循环图首先允许平衡状态，以及平衡状态的特征是什么是未知的。在这里，我们推导出规则，如何建立非循环，平面图允许从简单的循环图的平衡状态，通过添加环或边缘，从而识别不同类别的小平面网络允许平衡状态。研究平衡状态的特性，我们发现流域稳定性的方差与这些网络的图的大小成线性比例。我们引入平衡比作为一个新的序参量的基础上的流域稳定性的方法来分类网络上的平衡状态，并分析评估它的一个子集的网络类。我们的研究结果提供了一个分析描述的非循环图支持稳定，平衡的状态，从而可能有助于了解全局约束下的振荡器网络的拓扑要求。
英文摘要：
Phase balanced states are a highly under-explored class of solutions of the Kuramoto model and other coupled oscillator models on networks. So far, coupled oscillator research focused on phase synchronized solutions. Yet, global constraints on oscillators may forbid synchronized state, rendering phase balanced states as the relevant stable state. If for example oscillators are driving the contractions of a fluid filled volume, conservation of fluid volume constraints oscillators to balanced states as characterized by a vanishing Kuramoto order parameter. It has previously been shown that stable, balanced patterns in the Kuramoto model exist on circulant graphs. However, which non-circulant graphs first of all allow for balanced states and what characterizes the balanced states is unknown. Here, we derive rules of how to build non-circulant, planar graphs allowing for balanced states from the simple cycle graph by adding loops or edges to it. We thereby identify different classes of small planar networks allowing for balanced states. Investigating the balanced states' characteristics, we find that the variance in basin stability scales linearly with the size of the graph for these networks. We introduce the balancing ratio as a new order parameter based on the basin stability approach to classify balanced states on networks and evaluate it analytically for a subset of the network classes. Our results offer an analytical description of non-circulant graphs supporting stable, balanced states and may thereby help to understand the topological requirements on oscillator networks under global constraints.
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