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 concentrated near boundary regions.
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 identifies where the two spectral errors concentrate and is used to allocate supervision between prediction-level and representation-level alignment.
The deployed model remains a standard MLP and requires no graph access at inference.
Across node-classification benchmarks, G^2MLP consistently improves over graph-free distillation baselines, reduces the teacher-student rank gap in both regimes, and transfers without architectural changes to Graph Transformer teachers and link prediction.