Theoretical Foundations of Superhypergraph and Plithogenic Graph Neural Networks

TL;DR

This paper develops the theoretical foundations of Superhypergraph and Plithogenic Graph Neural Networks, extending message-passing principles to complex higher-order and multi-valued relational structures.

Contribution

It introduces rigorous definitions and fundamental properties for SHGNNs and Plithogenic GNNs, advancing the theoretical understanding of these complex neural network models.

Findings

Established structural properties of SHGNNs and Plithogenic GNNs

Proved well-definedness of key constructions in these models

Extended message-passing principles to higher-order structures

Abstract

Hypergraphs generalize classical graphs by allowing a single edge to connect multiple vertices, providing a natural language for modeling higher-order interactions. Superhypergraphs extend this paradigm further by accommodating nested, set-valued entities and relations, enabling the representation of hierarchical, multi-level structures beyond the expressive reach of ordinary graphs or hypergraphs. In parallel, neural networks-especially Graph Neural Networks (GNNs)-have become a standard tool for learning from relational data, and recent years have seen rapid progress on Hypergraph Neural Networks (HGNNs) and their theoretical properties. To model uncertainty and multi-aspect attributes in complex networks, several graded and multi-valued graph frameworks have been developed, including fuzzy graphs and neutrosophic graphs. The plithogenic graph framework unifies and refines these approaches by incorporating multi-valued attributes together with membership and contradiction mechanisms, offering a flexible representation for heterogeneous and partially inconsistent information. This book develops the theoretical foundations of SuperHyperGraph Neural Networks (SHGNNs) and Plithogenic Graph Neural Networks, with the goal of extending message-passing principles to these advanced higher-order structures. We provide rigorous definitions, establish fundamental structural properties, and prove well-definedness results for key constructions, with particular emphasis on strengthened formulations of Soft Graph Neural Networks and Rough Graph Neural Networks.