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.

翻译：超图通过允许单条边连接多个顶点，推广了经典图论，为建模高阶交互提供了自然的语言。超超图进一步扩展了这一范式，能够容纳嵌套的、集合值的实体与关系，从而可以表示超出普通图或超图表达能力的分层多级结构。与此同时，神经网络——尤其是图神经网络（GNNs）——已成为从关系数据中学习的标准工具，近年来超图神经网络（HGNNs）及其理论性质的研究取得了快速进展。为了对复杂网络中的不确定性和多属性方面进行建模，已发展出多种分级和多值图框架，包括模糊图和中性图。多值图框架通过整合多值属性以及隶属度与矛盾机制，统一并精炼了这些方法，为异构和部分不一致信息提供了灵活的表示。本书旨在建立超超图神经网络（SHGNNs）与多值图神经网络的理论基础，目标是将消息传递原理扩展到这些先进的高阶结构中。我们提供了严格的定义，建立了基本的结构性质，并证明了关键构造的良定义性结果，特别强调了软图神经网络与粗糙图神经网络的强化表述形式。