Abstract

Many real-world phenomena are naturally modeled by graphs and networks. However, classical graph models are often limited to pairwise interactions and may not adequately capture the richer structures that arise in practice. Higher-order graph formalisms extend this framework by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions, thereby providing more expressive representations of complex systems. This book presents a comprehensive overview of mathematical notions that can be used to model higher-order networks. It surveys foundational concepts, extensional frameworks, and newly introduced formalisms, with an emphasis on their structural principles, relationships, and modeling roles. The aim is to provide a unified perspective that helps readers compare diverse higher-order network models and identify appropriate tools for theoretical study and practical applications. This book is Edition 2.0. It mainly includes the addition of several concepts, as well as corrections and improvements of typographical errors and explanations. Keywords: Hypergraph, Superhypergraph, Higher-Order Graphs MSC2010 (Mathematics Subject Classification 2010): 05C65 - Hypergraphs, 05C82 - Graph theory with applications