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.

翻译：许多现实世界中的现象都自然地通过图和网络来建模。然而，经典图模型通常局限于成对交互，可能无法充分捕捉实践中出现的更丰富结构。高阶图形式通过引入多路、层次、时序、多层、递归和基于张量的交互来扩展这一框架，从而为复杂系统提供更具表达力的表示。本书全面概述了可用于建模高阶网络的数学概念，综述了基础概念、扩展框架及新引入的形式体系，重点强调了它们的结构原理、相互关系及建模角色。其目标是为读者提供一个统一的视角，帮助比较不同的高阶网络模型，并为理论研究和实际应用识别合适的工具。本书为2.0版，主要新增了若干概念，并对排版错误和解释进行了修正与改进。