Network data has become widespread, larger, and more complex over the years. Traditional network data is dyadic, capturing the relations among pairs of entities. With the need to model interactions among more than two entities, significant research has focused on higher-order networks and ways to represent, analyze, and learn from them. There are two main directions to studying higher-order networks. One direction has focused on capturing higher-order patterns in traditional (dyadic) graphs by changing the basic unit of study from nodes to small frequently observed subgraphs, called motifs. As most existing network data comes in the form of pairwise dyadic relationships, studying higher-order structures within such graphs may uncover new insights. The second direction aims to directly model higher-order interactions using new and more complex representations such as simplicial complexes or hypergraphs. Some of these models have long been proposed, but improvements in computational power and the advent of new computational techniques have increased their popularity. Our goal in this paper is to provide a succinct yet comprehensive summary of the advanced higher-order network analysis techniques. We provide a systematic review of its foundations and algorithms, along with use cases and applications of higher-order networks in various scientific domains.

翻译：随着时间推移，网络数据变得日益广泛、庞大且复杂。传统网络数据是二元的，仅捕捉成对实体间的关系。由于需要建模多于两个实体间的交互，大量研究聚焦于高阶网络及其表示、分析与学习方法。高阶网络研究主要分为两个方向：一是通过将基本研究单元从节点转变为称为模体的小型频繁观测子图，来捕捉传统（二元）图中的高阶模式——鉴于现有网络数据多呈现为成对二元关系，研究此类图中的高阶结构可能揭示新的洞见；二是利用单纯复形或超图等更新、更复杂的表示形式直接建模高阶交互。其中部分模型虽早已被提出，但计算能力的提升与新型计算技术的出现使其日益受到重视。本文旨在对先进的高阶网络分析技术进行简明而全面的总结，系统梳理其基础理论、算法以及在各科学领域中的用例与应用。