Automorphism-Induced Non-Canonicity in Top-k Explanations of Graph Neural Networks
Gradient-based GNN explainers assign equal attribution scores to chemically equivalent atoms (e.g., nitro groups) due to permutation equivariance, yet top-k explanations arbitrarily select one via array ordering. This is a structural obstruction: when no minimal explanation is fixed by the input’s automorphism group, no rule can be simultaneously single-valued, minimal, and symmetry-respecting. A parameter-free criterion in Lean 4 decides whether every score-optimal top-k report must split an or
Analysis
TL;DR
- Gradient-based GNN explainers assign equal attribution scores to chemically equivalent atoms (e.g., nitro groups) due to permutation equivariance, yet top-k explanations arbitrarily select one via array ordering.
- This is a structural obstruction: when no minimal explanation is fixed by the input’s automorphism group, no rule can be simultaneously single-valued, minimal, and symmetry-respecting.
- A parameter-free criterion in Lean 4 decides whether every score-optimal top-k report must split an orbit; validated across 21,298 instances with perfect agreement against mechanical model equivalence checks.
- Nontrivial automorphisms are pervasive (93.4% in Mutagenicity dataset), rendering measure-zero dismissals of symmetric inputs invalid in graph domains.
- Reporting orbits instead of individual nodes eliminates arbitrariness at negligible cost (0.11 ms, 0.43 extra edges per graph).
Why It Matters
This work exposes a fundamental flaw in how GNN explainers handle symmetric structures — a common occurrence in molecular graphs — revealing that current practices introduce arbitrary, unscientific choices under the guise of technical necessity. For AI practitioners and researchers in interpretability, it underscores the need for symmetry-aware explanation methods that respect inherent structural equivalences rather than breaking them via implementation artifacts like array indexing.
Technical Details
- Message-passing GNNs are permutation-equivariant, meaning automorphisms of the input graph leave all node/edge attributions invariant; thus, gradient-based assigners cannot distinguish between symmetric components.
- The paper formalizes the conflict between minimality, single-valuedness, and symmetry preservation in top-k explanations using group-theoretic reasoning on automorphism groups acting on explanation sets.
- A decision procedure implemented in Lean 4 determines, from graph structure alone, whether any optimal top-k explanation must split an orbit (i.e., select only part of a symmetric set); this was verified against exhaustive model-equivalence checks over 21,298 cases.
- Empirical analysis on MUTAG and Mutagenicity datasets shows that 24% of molecules with two interchangeable nitro groups have exactly one selected in top-k reports, chosen arbitrarily based on internal ordering.
- The proposed solution — reporting entire orbits instead of individual nodes — removes ambiguity while adding minimal overhead (average 0.43 extra edges per graph, 0.11 ms computation time).
Industry Insight
Explainability tools for graph-based models must explicitly account for symmetries inherent in their data domains (especially chemistry and biology); ignoring them leads to misleading or inconsistent interpretations that undermine trust and scientific validity. Adopting orbit-based reporting as a default for top-k explanations offers a lightweight, mathematically rigorous fix that preserves fidelity without sacrificing sparsity or performance, making it immediately deployable in production systems relying on GNN interpretability.
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