GegenbauerNet: Finding the Optimal Compromise in the GNN Flexibility-Stability Trade-off

TL;DR

GegenbauerNet introduces a spectral GNN filter that balances flexibility and stability by learning a single shape parameter, outperforming more adaptive models in local filtering tasks on heterophilic graphs.

Contribution

The paper proposes GegenbauerNet, a novel GNN filter based on Gegenbauer polynomials with a single learnable parameter, optimizing the trade-off between flexibility and stability in the spectral domain.

Findings

GegenbauerNet outperforms adaptive models in local filtering regimes.

Maximum flexibility benefits high-K filtering tasks.

Controlled symmetry and limited parameters improve stability and performance.

Abstract

Spectral Graph Neural Networks (GNNs) operating in the canonical [-1, 1] domain (like ChebyNet and its adaptive generalization, L-JacobiNet) face a fundamental Flexibility-Stability Trade-off. Our previous work revealed a critical puzzle: the 2-parameter adaptive L-JacobiNet often suffered from high variance and was surprisingly outperformed by the 0-parameter, stabilized-static S-JacobiNet. This suggested that stabilization was more critical than adaptation in this domain. In this paper, we propose \textbf{GegenbauerNet}, a novel GNN filter based on the Gegenbauer polynomials, to find the Optimal Compromise in this trade-off. By enforcing symmetry (alpha=beta) but allowing a single shape parameter (lambda) to be learned, GegenbauerNet limits flexibility (variance) while escaping the fixed bias of S-JacobiNet. We demonstrate that GegenbauerNet (1-parameter) achieves superior performance in the key local filtering regime (K=2 on heterophilic graphs) where overfitting is minimal, validating the hypothesis that a controlled, symmetric degree of freedom is optimal. Furthermore, our comprehensive K-ablation study across homophilic and heterophilic graphs, using 7 diverse datasets, clarifies the domain's behavior: the fully adaptive L-JacobiNet maintains the highest performance on high-K filtering tasks, showing the value of maximum flexibility when regularization is managed. This study provides crucial design principles for GNN developers, showing that in the [-1, 1] spectral domain, the optimal filter depends critically on the target locality (K) and the acceptable level of design bias.