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原文摘要

GNN-to-MLP distillation aims to retain the predictive accuracy of a message-passing teacher while deploying a graph-free MLP at inference. Existing methods mainly transfer node-wise predictions or use confidence-based reweighting, but they do not specify where the student should preserve the teacher's graph-induced geometry. We show that this omission leads to two spectral failure modes in the student's representation space. On sparse graphs, the student suffers from spectral underfit, missing high-energy teacher directions induced by aggregation. On dense graphs, it suffers from spectral overfit, retaining spurious directions that the teacher has collapsed through aggregation. Motivated by an energy-weighted teacher-student alignment objective, we propose Graph Geometry-aware MLP (G^2MLP), a training-time distillation framework guided by Ollivier-Ricci curvature. Curvature acts as a local proxy for allocating supervision between prediction-level and representation-level alignment. The deployed model is a standard MLP with no graph access at inference. Across node-classification benchmarks, G^2MLP improves over graph-free distillation baselines in most settings, reduces the teacher-student rank gap in both regimes, and transfers to Graph Transformer teachers and link prediction.