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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes

ICML 2025 · Graph learning › GNN architectures · approach: Physics-inspired GNNs · 0 citations (OpenAlex)

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Trains a directed GIN with edge features (DirGINE) to classify quiver mutation classes, uses PGExplainer and latent-space analysis to recover Vatne's type D characterization and independently discover and prove a characterization of type D-tilde quivers.

Problem

Characterizing quiver mutation equivalence classes (types A, D, E, affine A, D, E) in cluster algebras, where mutation equivalence is generally hard to decide and explicit criteria are known only for some types.

Approach

Train a GNN classifier on quivers of Dynkin and affine types, then use PGExplainer edge attributions and PCA/clustering of hidden activations to surface structural motifs, which guide a human-proved characterization of the D-tilde mutation class.

Key results

• DirGINE reaches 99.2% test accuracy on 11-node quivers (best epoch).
• On type D quivers, removing PGExplainer-highlighted edges flips 46.5% of predictions to type A, consistent with the known type D motifs.
• A linear classifier on layer-3 embeddings separates the type D subtypes with 99.7 ± 0.0% accuracy, and clustering guided a proved characterization of D-tilde quivers.

Measured effects reported in the paper

Surprising findings

• The model, trained only on type labels, separates the unlabeled type D subtypes linearly in its hidden space.
• Accuracy drops from 99.6% at n=12 to 89.1% at n=20, showing imperfect size generalization.

Limitations

• Fixed 4-layer depth may prevent recognizing long cycles and larger substructures, limiting size generalization.
• Explanations from PGExplainer are imperfect, with many ablated quivers flipping to type E, possibly out of distribution.
• Test set excludes type E-tilde and includes mutation-infinite E quivers, so accuracy is not a clean measure of classification.

Experimental setup

Datasets: quivers of types A, D, E, affine A, affine D, affine E generated with Sage

Benchmarks: held-out 11-node quivers

Metrics: classification accuracy cross-entropy loss edge-attribution masked prediction flip rates linear probe accuracy on latent embeddings

Builds on: PGExplainer graph isomorphism network Vatne classification of type D quivers

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