学习对称性的图形本质
Learning the Graphical Nature of Symmetries
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中文总结 AI 辅助
构建含131406个凯莱图的数据集研究有限群属性，给出新枚举贡献和经验规律。比较经典模型、MLP与图神经网络预测群属性，发现工程化图统计信息丰富，GNN能直接从图中恢复大量结构信号。
中文摘要 AI 辅助
有限群是刚性代数对象，其凯莱图展现出丰富网络几何结构，借此可度量、比较和学习群论结构。本文构建了包含131406个凯莱图的数据集，涵盖至多767阶（除512阶）的所有群，记录群属性的精确代数标签及多种图、循环、距离和谱统计信息。此普查旨在为研究有限群属性在凯莱图可观测量中的反映提供新基准，还给出新的枚举贡献。网络分析识别出若干经验规律并提出可测试猜想。最后比较了经典模型、MLP和图神经网络架构预测群属性的能力，结果表明工程化图统计信息丰富，GNN能直接从图中恢复大量结构信号。
英文摘要
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.
