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Beyond Oversquashing: Understanding Signal Propagation in GNNs Via Observables

NeurIPS 2026 · Graph learning › GNN architectures · approach: Structural GNN variants · 0 citations (OpenAlex)

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Proposes a quantum-inspired observable framework for measuring signal localization and routing on graphs, and a unitary spectral GNN (Schrödinger GNN) that uses feature-derivative Schrödinger Laplacians and complex modulation to route signals along learned feature directions.

Problem

Standard GNNs diffuse signals across the graph (oversmoothing/oversquashing) rather than routing them coherently between specific regions of interest; existing analyses do not measure targeted signal routing.

Approach

Model signal location, concentration, and routing using observables and their mean/variance (as in quantum mechanics), define a signal routing measure, and build a unitary Schrödinger propagator from feature-derivative Laplacians with complex feature modulation to induce feature-space momentum and directed transport.

Key results

• On the ring signal transport task, Schrödinger GNN reaches test loss 3e-04 versus 0.6644 for GCN and 0.6050 for GAT.
• On TU graph classification under architecture-matched evaluation, Schrödinger PMO achieves the best AP on all four datasets (e.g., 69.60 on IMDB vs 69.22 for GIN).
• On LRGB, Schrödinger GNN attains 0.7207 AP on Peptides-Func, outperforming all listed baselines (best prior 0.7173 Lie UniGCN).

Surprising findings

• Real-valued signals always have zero expected feature momentum, so complex modulation is necessary for directed routing.
• Standard spectral GNNs are shown to keep the signal's location unchanged and only increase its spread.

Limitations

• Applying the Schrödinger operator requires approximating the matrix exponential via repeated GSO applications, increasing cost relative to polynomial filters.
• Theoretical guarantees are developed only for Schrödinger Laplacian-based operators; general GSOs have fewer guarantees.
• The approach relies on the signal being localized in feature space, with localization analysis via window decomposition being conceptual.

Experimental setup

Datasets: ring graph synthetic signal transport ENZYMES IMDB MUTAG PROTEINS Roman-empire Amazon-ratings Minesweeper Tolokers Questions Peptides-Func Peptides-Struct

Benchmarks: TU datasets (ENZYMES, IMDB, MUTAG, PROTEINS) heterophilous node classification benchmarks LRGB (Peptides-Func, Peptides-Struct, PascalVOC-SP, COCO-SP) ring signal transport

Metrics: test AP ROC AUC test MAE test F1 test loss relative shift

Baselines: GCN GAT GIN UniGCN Lie UniGCN GUMP GraphSAGE GINE GatedGCN GPS DRew Exphormer

Builds on: GCN unitary graph convolutions (UniConv / Lie UniGCN) GUMP Chebyshev spectral networks

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