Superhypergraph Neural Networks and Plithogenic Graph Neural Networks: Theoretical Foundations
Takaaki Fujita
TL;DR
This work establishes a rigorous theoretical framework for SuperHyperGraph Neural Networks (SHGNNs) and Plithogenic Graph Neural Networks (PGNNs), positioning them as natural generalizations of Hypergraph Neural Networks (HGNNs) and classical GNNs while embracing uncertainty models. It builds a cohesive hierarchy linking HGNNs, F-GNNs, N-GNNs, and P-GNNs via n-SHGNN constructs, expanded hypergraphs, and spectral-like convolutions on base-vertex spaces. The paper provides formal definitions, Laplacians, convolution operators, and complexity analyses for SHGNN, $n$-SHGNN, and Dynamic/Multilevel variants, showing how HGNNs and familiar GNNs emerge as special cases. It further integrates dynamic, uncertain, and plithogenic viewpoints, illustrating how these frameworks can model complex real-world relations with degrees of appurtenance and contradiction. The theoretical foundations set the stage for future computational experiments and practical validations across domains requiring higher-order relationships and uncertainty handling.
Abstract
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.