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

2026-03-24

AI Summary

Real-world systems often exhibit complex interactions that transcend pairwise relationships, which conventional graph models struggle to capture effectively. This work presents a systematic review of graph-based modeling frameworks for higher-order networks, integrating diverse interaction paradigms—including simplicial, hierarchical, temporal, multilayer, recursive, and tensor-based representations—to clarify their structural principles and intrinsic connections. Building upon its first edition, this updated and comprehensive survey introduces key conceptual advances and refines theoretical formulations, thereby establishing a unified paradigm for higher-order network modeling. By offering a cohesive reference framework, the study significantly enhances the capacity to represent and analyze complex systems, supporting both theoretical development and practical applications in the field.

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.

Problem

• higher-order networks
• graph representation
• multiway interactions
• complex systems
• network modeling

Innovation

• higher-order networks
• graph-based frameworks
• multiway interactions
• tensor-based representations
• multilayer networks