This study addresses the limitations of the traditional analysis of message-passing, central to graph learning, by defining {\em \textbf{generalized propagation}} with directed and weighted graphs. The significance manifest in two ways. \textbf{Firstly}, we propose {\em Generalized Propagation Neural Networks} (\textbf{GPNNs}), a framework that unifies most propagation-based graph neural networks. By generating directed-weighted propagation graphs with adjacency function and connectivity function, GPNNs offer enhanced insights into attention mechanisms across various graph models. We delve into the trade-offs within the design space with empirical experiments and emphasize the crucial role of the adjacency function for model expressivity via theoretical analysis. \textbf{Secondly}, we propose the {\em Continuous Unified Ricci Curvature} (\textbf{CURC}), an extension of celebrated {\em Ollivier-Ricci Curvature} for directed and weighted graphs. Theoretically, we demonstrate that CURC possesses continuity, scale invariance, and a lower bound connection with the Dirichlet isoperimetric constant validating bottleneck analysis for GPNNs. We include a preliminary exploration of learned propagation patterns in datasets, a first in the field. We observe an intriguing ``{\em \textbf{decurve flow}}'' - a curvature reduction during training for models with learnable propagation, revealing the evolution of propagation over time and a deeper connection to over-smoothing and bottleneck trade-off.

本研究针对传统的消息传递分析在图学习中的局限性，通过定义具有定向性和加权图的“广义传播”，提出了广义传播神经网络（GPNNs）框架来统一大部分基于传播的图神经网络，并对设计空间中的权衡进行了实验验证，并通过理论分析强调了邻接函数对模型表达能力的关键作用；其次，提出了连续统一Ricci曲率（CURC），用于有向和加权图，通过理论证明CURC具有连续性、尺度不变性，并与Dirichlet等面积常数保持较低的连接，从而验证了GPNNs的狭口分析。在数据集中对学习到的传播模式进行了初步探索，首次在该领域进行了实证研究。我们观察到一种有趣的“曲率减小流”，即在具有可学习传播的模型训练过程中，曲率逐渐降低，揭示了传播随时间的演变以及与过度平滑和狭口权衡之间的深层联系。

揭示广义图传播的弯曲流程