用拓扑方法分析图神经网络的特征传播，揭示节点与边的局部一致性机制。
A Sheaf-Theoretic and Topological Perspective on Complex Network Modeling and Attention Mechanisms in Graph Neural Models
- 基于细胞层化理论建模节点特征与边权重的一致性
- 通过多尺度拓扑分析捕捉层级特征交互关系
- 适用于图学习中的节点分类与社区发现任务
组合与拓扑结构（如图、单纯复形、胞腔复形）是几何与拓扑深度学习（GDL和TDL）架构的基础。这些模型在上述域上聚合信号、整合局部特征，并生成用于多种现实应用的表示。然而，GDL和TDL特征在训练过程中的分布与扩散行为仍是一个开放且未被充分探索的问题。针对这一空白，本文提出一种细胞层化理论框架，用于建模和分析基于图架构中节点特征与边权重的局部一致性和调和性。通过跟踪层化结构中的局部特征对齐与共识，该框架为特征扩散与聚合提供了拓扑视角。此外，受拓扑数据分析（TDA）启发，提出了一个多层次扩展，以捕捉图模型中的层次化特征交互。该方法基于底层几何与拓扑结构以及其上学习信号的联合表征，为节点分类、子结构检测和社区检测等传统任务提供了新洞察。
原文摘要 · Abstract (English)
Combinatorial and topological structures, such as graphs, simplicial complexes, and cell complexes, form the foundation of geometric and topological deep learning (GDL and TDL) architectures. These models aggregate signals over such domains, integrate local features, and generate representations for diverse real-world applications. However, the distribution and diffusion behavior of GDL and TDL features during training remains an open and underexplored problem. Motivated by this gap, we introduce a cellular sheaf theoretic framework for modeling and analyzing the local consistency and harmonicity of node features and edge weights in graph-based architectures. By tracking local feature alignments and agreements through sheaf structures, the framework offers a topological perspective on feature diffusion and aggregation. Furthermore, a multiscale extension inspired by topological data analysis (TDA) is proposed to capture hierarchical feature interactions in graph models. This approach enables a joint characterization of GDL and TDL architectures based on their underlying geometric and topological structures and the learned signals defined on them, providing insights for future studies on conventional tasks such as node classification, substructure detection, and community detection.
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