Edge-Level Automorphism in GNNs: A Quantitative Framework and Effective Designs For Link Prediction
Chen Shao, Donald Loveland, Tobias Käfer, Danai Koutra
Abstract
Graph Neural Networks (GNNs) are effective for learning node and link embeddings through permutation-equivariant aggregation. However, standard GNNs collapse automorphic nodes, i.e., those with identical structural roles (or orbits) into indistinguishable representations, leading to the node automorphism problem. This collapse limits their expressive power and degrades link prediction performance. Existing approaches to characterize GNN expressiveness rely primarily on Weisfeiler-Lehman (WL) analyses, but these methods are typically qualitative and often misaligned with empirical results. To address this gap, we begin by introducing a novel quantitative framework to assess GNN expressiveness for link prediction. We first formalize edge-level automorphism through edge orbits, which capture the set of structural role pairs for nodes that share a link. Then, we introduce the edge automorphism ratio (EAR), a scalar metric that quantifies a GNN's ability to distinguish links in a given graph. We empirically demonstrate that EAR correlates strongly with performance, validating its practical benefit. Building on this insight, we design EDGE-ORBIT EQUIVARIANT GRAPH NEURAL NETWORK (EO-GNN), a GNN architecture that addresses automorphism collapse while preserving equivariance and incurring minimal computational overhead. EO-GNN accomplishes this through two core designs combined with WL-based node hashes: (i) automorphism-aware dropouts and (ii) subgraph orbit-biased aggregation. Empirical evaluations on synthetic and real graphs show improvements of up to 42.36% and 28.44%, respectively, in predicting links in scenarios with high automorphism.