图的Forman-Ricci曲率的马尔可夫基与格基
Markov and lattice bases for Forman-Ricci curvature of graphs
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中文总结 AI 辅助
本研究发展图的Forman-Ricci曲率相关马尔可夫基的代数组合理论，证明不可或缺马尔可夫移动度数随图最大度数二次增长，用三度马尔可夫移动构造格基并结合强化学习寻找特定图的马尔可夫移动。
中文摘要 AI 辅助
离散Forman-Ricci曲率是与图每条边相关联、描述其局部几何的量，已被证明是多种应用中网络分析的有用工具。Roost等人（2024）近期的研究提出使用马尔可夫基从具有给定顶点度和曲率的图空间中采样。本研究进一步发展这些马尔可夫基的代数与组合理论，证明不可或缺的马尔可夫移动的度数至少随图的最大度数呈二次增长；鉴于该结果，目前似乎无法实现所有马尔可夫基元素的紧凑描述，转而仅使用三度马尔可夫移动为该问题找到格基，这使我们能够采用最新开发的强化学习方法寻找可应用于特定图的马尔可夫移动。
英文摘要
Discrete Forman-Ricci curvature is a quantity associated to each edge of a graph that describes its local geometry. It has proven to be a useful tool in network analysis in a variety of applications. Recent work by Roost et al.\ (2024) proposed the use of Markov bases to sample from the space of graphs with prescribed vertex degrees and curvatures. In the present work, we further develop the algebraic and combinatorial theory of these Markov bases. We show that the degree of an indispensable Markov move grows at least quadratically in the maximum degree of the graph. In light of this result, a compact description of all Markov basis elements seems unattainable at present. Instead, we find a lattice basis for this problem using only degree three Markov moves, which allows us to employ recently-developed reinforcement learning methods for finding Markov moves that can be applied to a specific graph.
