Superhypergraph Neural Networks and Plithogenic Graph Neural Networks: Theoretical Foundations

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.