arXiv:2511.13730v1 Announce Type: cross 
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.

تقدم الورقة GegenbauerNet، وهو نوع جديد من الشبكات العصبية الطيفية (GNN) مصمم لمعالجة التوازن بين المرونة والاستقرار الذي يتسم به GNNs التي تعمل ضمن المجال [-1، 1]. أظهرت الأبحاث السابقة أن L-JacobiNet التكيفي غالبًا ما يعاني من تباين مرتفع وتفوق عليه S-JacobiNet المستقر. يستخدم GegenbauerNet كثيرات حدود غينغنباور، مما يفرض التناظر مع السماح بتعلم معلمة شكل واحدة، مما يحد من التباين ويحسن الأداء في مهام التصفية المحلية.

El artículo presenta GegenbauerNet, un nuevo tipo de Red Neuronal Gráfica Espectral (GNN) diseñada para abordar el compromiso entre Flexibilidad y Estabilidad inherente a las GNN que operan en el dominio [-1, 1]. Investigaciones anteriores indicaron que el L-JacobiNet adaptativo a menudo mostraba alta varianza y era superado por el S-JacobiNet estabilizado. GegenbauerNet utiliza polinomios de Gegenbauer, imponiendo simetría mientras permite aprender un solo parámetro de forma, limitando así la varianza y mejorando el rendimiento en tareas de filtrado local.

Cet article présente GegenbauerNet, un nouveau type de réseau de neurones graphes spectral (GNN) conçu pour traiter le compromis Flexibilité-Stabilité inhérent aux GNN opérant dans le domaine [-1, 1]. Des recherches antérieures ont montré que le L-JacobiNet adaptatif présentait souvent une forte variance et était surpassé par le S-JacobiNet stabilisé. GegenbauerNet utilise des polynômes de Gegenbauer, imposant une symétrie tout en permettant l'apprentissage d'un seul paramètre de forme, limitant ainsi la variance et améliorant les performances dans les tâches de filtrage local.

The paper introduces GegenbauerNet, a new type of Spectral Graph Neural Network (GNN) designed to address the Flexibility-Stability Trade-off inherent in GNNs operating within the [-1, 1] domain. Previous research indicated that the adaptive L-JacobiNet often exhibited high variance and was outperformed by the stabilized S-JacobiNet. GegenbauerNet employs Gegenbauer polynomials, enforcing symmetry while allowing a single shape parameter to be learned, thus limiting variance and improving performance in local filtering tasks.

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

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