Endpoint Symmetry for Edge Updates: Weight-Space Redundancy in GNNs on Undirected Graphs

Charlotte Cambier van Nooten, Stijn van den Beemt, Yuliya Shapovalova, Tom Heskes

Keywords: Graph Neural Networks, Deep Sets, Edge Classification, Endpoint Symmetry, Weight-Space Symmetry

Abstract: Graph Neural Networks (GNNs) for undirected graphs typically update edge representations by concatenating endpoint embeddings. Because an undirected edge $\{u,v\}$ has no canonical ordering, concatenation violates endpoint symmetry: the output depends on an arbitrary serialisation choice, not on the graph. The model must spend capacity learning a symmetry it cannot structurally enforce, with no guarantee of finding it. For any weight configuration $\theta$, a partner $\theta'$ obtained by permuting the $h_u$ and $h_v$ input blocks of the first weight matrix produces the same model prediction on every undirected edge, inducing a redundant $S_2$ orbit in weight space so different seeds may land in either half. We resolve this with a Deep Sets aggregation $\rho(g, \phi(h_u) + \phi(h_v))$ that enforces endpoint symmetry by construction, collapses the orbit, and approximates any continuous endpoint-symmetric function arbitrarily well on compact domains. Experiments confirm three predictions: different outputs for $(u,v)$ and $(v,u)$, geometrically distinct seed configurations, and a pronounced interpolation barrier. A real-world power grid application yields higher recall, lower variance, and one-third fewer parameters. Matching architecture symmetry to data symmetry eliminates redundant orbits at no cost to expressivity.

Email Sharing: We authorize the sharing of all author emails with Program Chairs.

Data Release: We authorize the release of our submission and author names to the public in the event of acceptance.

Submission Number: 21