通过图嵌入的广义斜对称复制方程的可积性
Integrability of Generalised Skew-Symmetric Replicator Equations via Graph Embeddings
AI总结：
本文研究有向图与零和复制动力学之间的可积性关系，证明图嵌入保持 Liouville-Arnold 可积性，构造新的可积图族，并对六顶点有向图进行分类，发现三个不属于已知分类的保留图，提示图组合结构与可积动力学之间可能存在深层联系。
摘要：
已知有向定向图与零和复制动力学（Lotka-Volterra 方程）之间存在一一映射，并且这些动力学在适当定义的非线性 Poisson 括号中是哈密顿的。本文研究确定这些动力学是否 Liouville-Arnold 可积的问题，建立在 Evripidou 等人 [J. Phys. A., 55:325201, 2022] 的图去克隆（graph decloning）以及 Paik 和 Griffin [Phys. Rev. E. 107(5): L052202, 2024] 的图嵌入（graph embedding）等先前工作之上。利用 Paik 和 Griffin 的嵌入程序，我们证明（在一定的限制条件下）当一个产生可积动力学的图被嵌入到另一个产生可积动力学的图中时，所得的图结构也产生可积动力学。我们还构造了一个新的图结构族，它产生可积动力学，但既不来自嵌入也不来自去克隆。我们利用这些结果以及数值方法，对六个顶点上几乎所有有向定向图所产生的动力学进行分类，其中有三个保留图（hold-out graphs）产生可积动力学，且不属于由已知族和图运算产生的自然分类体系。这些保留图表明还有更多结构有待发现。此外，该工作表明，导致可积动力学的有向定向图或许可以以类似于有限单群分类的方式进行分类，从而创造了一种可能性，即图的组合结构与可积动力学之间存在深层联系。
英文摘要：
It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville-Arnold integrable, building on prior work graph in graph decloning by Evripidou et al. [J. Phys. A., 55:325201, 2022] and graph embedding by Paik and Griffin [Phys. Rev. E. 107(5): L052202, 2024]. Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results, along with numerical methods, to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs.
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