Hypergraphs extend traditional graphs by allowing edges to connect multiple nodes, while superhypergraphs further generalize this concept to represent even more complex relationships. Neural networks, inspired by biological systems, are widely used for tasks such as pattern recognition, data classification, and prediction. Graph Neural Networks (GNNs), a well-established framework, have recently been extended to Hypergraph Neural Networks (HGNNs), with their properties and applications being actively studied. The Plithogenic Graph framework enhances graph representations by integrating multi-valued attributes, as well as membership and contradiction functions, enabling the detailed modeling of complex relationships. In the context of handling uncertainty, concepts such as Fuzzy Graphs and Neutrosophic Graphs have gained prominence. It is well established that Plithogenic Graphs serve as a generalization of both Fuzzy Graphs and Neutrosophic Graphs. Furthermore, the Fuzzy Graph Neural Network has been proposed and is an active area of research. This paper establishes the theoretical foundation for the development of SuperHyperGraph Neural Networks (SHGNNs) and Plithogenic Graph Neural Networks, expanding the applicability of neural networks to these advanced graph structures. While mathematical generalizations and proofs are presented, future computational experiments are anticipated.

翻译：超图通过允许边连接多个节点扩展了传统图，而超超图进一步推广了这一概念以表示更复杂的关系。受生物系统启发的神经网络被广泛用于模式识别、数据分类和预测等任务。图神经网络（GNNs）作为一个成熟框架，最近已扩展至超图神经网络（HGNNs），其性质和应用正被积极研究。Plithogenic图框架通过整合多值属性以及隶属度和矛盾函数，增强了图的表示能力，从而能够对复杂关系进行详细建模。在处理不确定性的背景下，模糊图和中性图等概念已变得日益重要。学界公认Plithogenic图是模糊图和中性图的泛化。此外，模糊图神经网络已被提出并成为活跃的研究领域。本文为超超图神经网络（SHGNNs）和Plithogenic图神经网络的发展奠定了理论基础，将神经网络的适用性扩展至这些先进的图结构。文中提出了数学泛化与证明，并展望了未来的计算实验。