边围长：图神经网络的一种结构边特征
Edge-Girth as a Structural Edge Feature for Graph Neural Networks
- LTCI, Télécom Paris, Institut Polytechnique de Paris（LTCI、巴黎电信学院、巴黎综合理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出将边围长及其重数作为结构边特征，构建EGAGNN模型，在ZINC-12k回归基准上表现优于对比模型，但在BREC基准的特定图对上无法突破1-WL界限。
AI中文摘要：
基于消息传递的图神经网络（GNN）被证明其能力不超过一维Weisfeiler-Leman颜色细化测试（1-WL）：无论网络深度和宽度如何，它无法区分的两个图会得到相同的表示。常见的解决方法是用预计算的结构描述符增强节点或边特征，最常见的是固定小子图（如三角形或更长的环）的数量，但这类计数需要预先确定所计数子结构的大小，该选择通常是在不考虑数据的情况下做出的。本文研究一种避免此选择的描述符：边的边围长是经过该边的最短环的长度，其重数是此类最短环的数量；二者共同构成一种逐边不变量，可报告任意长度的环，且每条边仅需一次广度优先搜索即可精确计算。将其注入门控消息传递架构EGAGNN后，在ZINC-12k回归基准上，参数为10.4万时，测试MAE比最接近的门控对比模型低3倍；在相同架构下，与有界环计数描述符相比，它仅与计数长度不超过8的环的字典表现相当，但使用了两倍的通道数，而长度不超过4的环的字典表现与完全不使用结构信息无差异。在图判别任务上，我们证明了匹配的局限性：在每条边看到相同数量、相同长度最短环的图上，该描述符会变为常数，基于它构建的任何模型都会退回到1-WL的界限。这在BREC基准的全部400对图中无一例外：90对此类图中没有一对能被区分。
英文摘要：
Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1-WL): two graphs it cannot tell apart receive identical representations, however deep or wide the network. A common remedy augments node or edge features with precomputed structural descriptors, most often counts of a fixed small subgraph such as triangles or longer cycles, but such counts require committing in advance to the size of the substructure counted, a choice usually made blind to the data. We study a descriptor that avoids this choice. The edge-girth of an edge is the length of a shortest cycle through it, and its multiplicity is the number of such shortest cycles; together they form a per-edge invariant that reports cycles of arbitrary length, computable exactly by a single breadth-first search per edge. Injected into a gated message-passing architecture, EGAGNN, it reaches a test MAE a factor three below the closest gated comparator on the ZINC-12k regression benchmark at 104k parameters; against bounded cycle-counting descriptors under the same architecture, it matches only a dictionary counting cycles up to length eight, using twice as many channels, while a dictionary capped at length four performs no better than no structural information at all. On graph discrimination we prove a matching limitation: on graphs where every edge sees the same number of shortest cycles of the same length, the descriptor becomes constant and any model built on it collapses back to the 1-WL bound. This holds without exception across all 400 pairs of the BREC benchmark: not one of the 90 such pairs is distinguished.
