Representing Higher-Order Networks: A Survey of Graph-Based Frameworks

Many real-world phenomena can be naturally represented using graphs and networks. Classical graph models, however, are generally restricted to pairwise interactions and may therefore be insufficient for capturing the richer structural relationships encountered in practical systems. Higher-order graph formalisms extend the classical framework by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions, thereby enabling more expressive representations of complex systems. This book provides a comprehensive overview of mathematical concepts and frameworks for modeling higher-order networks. It surveys foundational notions, extended frameworks, and newly introduced formalisms, with particular emphasis on their underlying structural principles, interrelationships, and respective modeling roles. Its primary objective is to offer a unified perspective that enables readers to compare diverse higher-order network models and to identify suitable mathematical tools for both theoretical investigation and practical application. Now in its fourth edition, this volume presents an expanded, revised, and corrected treatment of foundational concepts, established extensions, and recently introduced structures, while maintaining a unifying focus on the relationships among these models. It is intended to serve as a reference for researchers and students studying and comparing higher-order network models in combinatorics, topology, applied mathematics, and related fields.