构建13万张群的凯莱图，揭示代数结构与图几何的关联。
Learning the Graphical Nature of Symmetries
- 构建覆盖13万张凯莱图的数据集，含精确代数标签与图统计量。
- 发现尼尔森群的谱特征、直径与聚类间存在可验证规律。
- 图神经网络能从图结构中直接学习群性质，表现优于传统模型。
有限群是刚性的代数对象，其凯莱图展现出丰富的网络几何结构，可用于度量、比较和学习群的代数特性。本文构建了包含131,406张凯莱图的数据集，覆盖所有阶数不超过767的群（512阶除外），记录了精确的代数标签以及广泛的图、环、距离和谱统计量。该普查旨在为研究有限群性质如何反映在凯莱图可观测量上提供新基准。同时产出新的枚举结果：在恢复标准群类已知OEIS序列的基础上，新增了单块群及由最多三、四、五个元素生成的群的新序列至OEIS。伴随的网络分析识别出若干经验规律，并提出可检验的猜想，包括平方聚类、凯莱图直径、平均图紊乱度与幂零群的谱特征间隙之间的关系。最后，对经典模型、MLP及图神经网络架构进行了对比实验，以直接从凯莱图数据预测代数群性质。结果显示，人工设计的图统计量极具信息量，而图神经网络（尤其是GIN，在某些固定阶设置下为GCN）能直接从图中恢复大量结构信号，表明图感知架构在这些群论图表示上表现出最优阶段。
原文摘要 · Abstract (English)
Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.
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