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

arXiv:2511.17367v1 Announce Type: new 
Abstract: Computing worst-case robust strategies in pursuit-evasion games (PEGs) is time-consuming, especially when real-world factors like partial observability are considered. While important for general security purposes, real-time applicable pursuit strategies for graph-based PEGs are currently missing when the pursuers only have imperfect information about the evader's position. Although state-of-the-art reinforcement learning (RL) methods like Equilibrium Policy Generalization (EPG) and Grasper provide guidelines for learning graph neural network (GNN) policies robust to different game dynamics, they are restricted to the scenario of perfect information and do not take into account the possible case where the evader can predict the pursuers' actions. This paper introduces the first approach to worst-case robust real-time pursuit strategies (R2PS) under partial observability. We first prove that a traditional dynamic programming (DP) algorithm for solving Markov PEGs maintains optimality under the asynchronous moves by the evader. Then, we propose a belief preservation mechanism about the evader's possible positions, extending the DP pursuit strategies to a partially observable setting. Finally, we embed the belief preservation into the state-of-the-art EPG framework to finish our R2PS learning scheme, which leads to a real-time pursuer policy through cross-graph reinforcement learning against the asynchronous-move DP evasion strategies. After reinforcement learning, our policy achieves robust zero-shot generalization to unseen real-world graph structures and consistently outperforms the policy directly trained on the test graphs by the existing game RL approach.

يقدم مقال جديد استراتيجيات مطاردة قوية في الوقت الحقيقي (R2PS) لألعاب المطاردة والهروب (PEGs) في ظل ظروف الرؤية الجزئية. يتناول هذا النهج التحدي المتمثل في تطوير استراتيجيات مطاردة فعالة عندما يكون لدى المطاردين معلومات محدودة عن موقع الهارب، وهو نقص كبير في الأبحاث الحالية.

Un nuevo artículo introduce estrategias de persecución robustas en tiempo real (R2PS) para juegos de persecución-esquiva (PEGs) bajo condiciones de observabilidad parcial. Este enfoque aborda el desafío de desarrollar estrategias de persecución efectivas cuando los perseguidores tienen información limitada sobre la posición del evasor, una brecha significativa en la investigación actual.

Un nouvel article présente des stratégies de poursuite robustes en temps réel (R2PS) pour les jeux de poursuite-évasion (PEGs) dans des conditions d'observabilité partielle. Cette approche répond au défi de développer des stratégies de poursuite efficaces lorsque les poursuivants disposent d'informations limitées sur la position de l'évadé, un manque significatif dans la recherche actuelle.

A new paper introduces worst-case robust real-time pursuit strategies (R2PS) for pursuit-evasion games (PEGs) under conditions of partial observability. This approach addresses the challenge of developing effective pursuit strategies when pursuers have limited information about the evader's position, a significant gap in current research.

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

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