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

TL;DR

This work examines the Flexibility-Stability Trade-off in spectral GNNs within the canonical $[-1,1]$ domain, showing that highly adaptive filters can overfit while overly stabilized filters may be biased. It introduces GegenbauerNet, a 1-parameter, symmetric AOPF based on Gegenbauer polynomials, to achieve an Optimal Compromise between bias and variance. Through extensive experiments across seven datasets covering both homophily and heterophily, the study reveals that GegenbauerNet excels in local filtering regimes ($K=2$) on heterophilic graphs, while highly flexible $L$-Jacobi nets perform best on several homophilic tasks, and that fixed-bias $S$-JacobiNet offers strong regularization in high-$K$ regimes. The results provide actionable guidelines: use $S$-JacobiNet for high-$K$ global filtering, adopt GegenbauerNet for low-$K$ local filtering, and consider the full AOPF-Jacobi family to tailor spectral filters to target locality and bias tolerance.

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.

Figures (3)

• Figure 1: Training Dynamics Comparison ($K=3, H=16$). On homophilic datasets, all models are competitive. On heterophilic datasets (Texas, Cornell, Wisconsin, Chameleon), the AOPF-Jacobi family (S, L, Gegenbauer) converges stably to significantly higher accuracy than GAT/APPNP, confirming the superiority of spectral adaptation over spatial methods in these challenging regimes.
• Figure 2: $K$ (Polynomial Degree) vs. Test Accuracy on PubMed (Homophilic). LJacobiNet (2-param) is the most robust, stabilizing at $\sim 0.72$ at $K=10$, while SJacobiNet (0-param) collapses from lack of filter expressiveness.
• Figure 3: $K$ (Polynomial Degree) vs. Test Accuracy on Texas (Heterophilic, 10-fold Avg). GegenbauerNet (1-param) defines the optimal compromise at the local setting ($K=2$), achieving the highest average accuracy. However, SJacobiNet (0-param) surprisingly wins at higher degrees ($K=5, 10$), demonstrating the beneficial regularization of a fixed, hard constraint for global feature mixing.