Edge-Girth as a Structural Edge Feature for Graph Neural Networks

Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1-WL): starting from a uniform colouring, each node is repeatedly recoloured as a function of its own colour and the multiset of its neighbours' colours, and two graphs the process 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 being 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 and is computable exactly by a single breadth-first search per edge. Like other structural encodings, it is computed once from the graph alone, before and independently of any supervision, and is therefore not specific to a downstream objective. We evaluate it on two tasks of different kinds. 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 $104$k parameters; compared against bounded cycle-counting descriptors under the same architecture, it matches only a dictionary that explicitly counts cycles up to length eight, which requires 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 prediction holds without exception across all 400 pairs of the BREC graph-isomorphism benchmark: not one of the 90 such pairs is distinguished.