Spectral Graph Neural Networks with Hermite Polynomials: A Comprehensive Study

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We study spectral graph neural networks built from Hermite polynomials and propose HermNet, a simple model that combines a nodewise predictor with normalized Hermite propagation. Its sparse recurrence requires neither eigendecomposition nor a learned basis. We distinguish the basic model from optional coordinate calibration, response normalization and Gaussian derivative regularization. Hermite and other complete polynomial bases span the same degree-bounded filter space, but their coordinates can produce different optimization behavior under limited training budgets. We analyze this behavior through spectral signal energy, label sampling, changes in learned features and the bias--variance trade-off of regularization. Controlled synthetic experiments identify a regime in which plain HermNet outperforms matched polynomial-basis alternatives, including with a jointly trained nonlinear predictor. Curvature regularization further improves HermNet when the same functional penalty is available to every comparator. Fixed-predictor controls support the advantage under short training budgets, but longer training removes the plain-model lead. Matched real-data comparisons show accuracy deficits, and architectural and numerical studies identify further limits. Together, the analysis and experiments clarify when Hermite propagation is useful and how calibration and regularization affect its performance.