通过通用层方程统一图神经网络

摘要

图神经网络通常通过面向特定家族的方程来描述，这些方程的符号表示掩盖了共享计算与结构差异。我们引入了一个通用层方程，通过七个组件来表示所涵盖的架构：更新域、通道集、传播库、逐通道消息映射、通道融合算子、自身/残差映射以及更新映射。该核心分解将信息传递的位置（由传播库编码）与信息传递的内容（由消息映射编码）分离开来。函数值填充将同一方程扩展到局部消息传递、注意力机制、谱滤波、全局通信、关系特定通道、高阶域和几何消息之中。我们通过规范层的详细化简和跨越七个非互斥架构家族的组件分配，使这种统一性明确且可检验。固定的槽位规则根据计算角色分配操作，并界定了框架的覆盖范围。该分解还产生了组件层面的理论洞见：在端点局部消息和节点局部更新下，算子支撑集限定了单层依赖范围；而在所述假设下，单层全局混合需要算子有效行为满行。由此产生的框架在公共设计空间中组织了200多种架构，支持逐组件比较以及结构一致架构的生成，并将传播选择与过平滑、过挤压、异质性及表达能力联系起来。它还揭示了这样一个实证逆问题：如何将可测量的图与任务属性映射到经过验证的组件选择上。

English

Graph neural networks are commonly described through family-specific equations whose notation obscures shared computations and structural differences. We introduce a common layer equation that represents covered architectures through seven components: an update domain, channel set, propagation bank, per-channel message maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by the propagation bank, from what moves, encoded by the message maps. Function-valued fillings extend the same equation across local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. We make this unification explicit and checkable through worked reductions of canonical layers and component assignments spanning seven nonexclusive architectural families. A fixed slot discipline assigns operations by computational role and defines the framework's coverage boundary. The decomposition also yields component-level theoretical insights: under endpoint-local messages and node-local updates, operator support bounds one-layer dependencies, and one-layer global mixing requires a full effective operator row under the stated hypotheses. The resulting framework organizes more than 200 architectures in a common design space, enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.