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Graph Neural Networks for Edge Signals: Orientation Equivariance and Invariance
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Proposes joint orientation equivariance/invariance desiderata and EIGN, a direction-aware magnetic-Laplacian GNN for edge-level signals that handles both orientation-equivariant and -invariant features on graphs with directed and undirected edges.
Problem
Edge-level prediction on graphs where edge signals may be orientation-equivariant (directional, e.g. flow) or orientation-invariant (e.g. diameter), and edges may be directed or undirected; existing topological and line-graph GNNs cannot handle both or distinguish edge direction.
Approach
Defines joint orientation equivariance/invariance restricted to undirected edges, and builds EIGN from magnetic edge Laplacians (complex phase shifts for directed edges) for equivariant and invariant signals, plus Laplacian-based and pointwise fusion operators between the two modalities.
Key results
- On real-world simulation tasks, EIGN reduces RMSE by up to 23.5% relative to baselines.
- On Tri-Flow synthetic task, EIGN reaches 0.864/0.996/0.022 versus runner-up around 0.819/0.799/0.293 (per table).
- Ablating direction-awareness (q=0) and fusion operators degrades performance on synthetic and real data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use direction-aware complex-valued Laplacians instead of the direction-agnostic variant (q=0) (ablation) | EIGN · Random Walk Completion (RW Comp) | performance | Helped |
| Use direction-aware complex-valued Laplacians instead of the direction-agnostic variant (q=0) (ablation) | EIGN · Longest Directed Cycle Prediction (LD Cycles) | performance | Helped |
| Use direction-aware complex-valued Laplacians instead of the direction-agnostic variant (q=0) (ablation) | EIGN · Real-world flow simulation | RMSE | Helped |
| Include Laplacian fusion operators between equivariant and invariant signals (ablation) | EIGN · Simulation tasks without orientation-equivariant features | performance | Helped |
| Include the same-edge fusion operation (ablation) | EIGN · Triangle Flow Orientation (Tri-Flow) | performance | Helped |
| Include the same-edge fusion operation (ablation) | EIGN · Real-world flow tasks | performance | Helped |
| Use EIGN rather than prior methods | Edge-level synthetic tasks · Random Walk Completion (RW Comp) | performance | Helped |
| Use EIGN rather than prior methods | Edge-level synthetic tasks · Longest Directed Cycle Prediction (LD Cycles) | performance | Helped |
| Use EIGN rather than prior methods | Edge-level synthetic tasks · Triangle Flow Orientation (Tri-Flow) | performance | Helped |
| Use EIGN rather than baselines | Real-world flow simulation · Traffic networks and electrical circuits | RMSE: up to 23.5% | Helped |
Surprising findings
- Direction-agnostic baselines such as HodgeGNN and line-graph GNNs perform poorly when inputs mix orientation-equivariant and invariant signals.
Limitations
- Only edge-level problems studied; higher-order topological structures not modeled.
- Local message passing only; global propagation left for future work.
- Magnetic Laplacian is one possible framework; other direction-aware approaches may work as well.
Experimental setup
Datasets: Anaheim Barcelona Chicago Winnipeg RW Comp (synthetic) LD Cycles (synthetic) Tri-Flow (synthetic) LTSpice electrical circuits
Benchmarks: RW Comp LD Cycles Tri-Flow Anaheim Barcelona Chicago Winnipeg electric circuits
Metrics: accuracy RMSE
Baselines: MLP LineGraphGNN HodgeGNN Hodge+Inv Hodge+Dir Line-MagNet Dir-GNN
Builds on: HodgeGNN Magnetic Node Laplacian Dir-GNN Line graph GNN
Closest papers by meaning
- Topology-aware Neural Flux Prediction Guided by Physics (ICML 2025 · similarity 0.86)
- Improving Graph Neural Networks by Learning Continuous Edge Directions (ICLR 2025 · similarity 0.86)
- From Trainable Negative Depth to Edge Heterophily in Graphs (NeurIPS 2023 · similarity 0.84)
- E(n) Equivariant Topological Neural Networks (ICLR 2025 · similarity 0.84)
- The Merit of River Network Topology for Neural Flood Forecasting (ICML 2024 · similarity 0.83)
- What Is Missing For Graph Homophily? Disentangling Graph Homophily For Graph Neural Networks (NeurIPS 2024 · similarity 0.83)
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