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复杂网络是描述和理解现实世界中复杂系统的有力工具. 近年来, 为了更准确地描述复杂网络中的交互关系, 或者从高阶视角分析成对交互作用网络, 许多学者开始使用高阶网络进行建模, 并在研究其动力学过程中发现了与成对交互作用网络不同的新现象. 然而, 与成对交互作用网络相比, 高阶网络的研究相对较少; 而且, 高阶网络结构相对复杂, 基于结构的统计指标定义较为分散且形式不统一, 这些都给描述高阶网络的拓扑结构特征带来了困难. 鉴于此, 本文综述了两种最常见的高阶网络——超图和单纯形网络——常用的统计指标及其物理意义. 本文有助于加深对高阶网络的理解, 促进对高阶网络结构特征的定量化研究, 也有助于研究者在此基础上开发更多适用于高阶网络的统计指标.Complex networks serve as indispensable instruments for characterizing and understanding intricate real-world systems. Recently, researchers have delved into the realm of higher-order networks, seeking to delineate interactions within these networks with greater precision or analyze traditional pairwise networks from a higher-dimensional perspective. This effort has unearthed some new phenomena different from those observed in the traditional pairwise networks. However, despite the importance of higher-order networks, research in this area is still in its infancy. In addition, the complexity of higher-order interactions and the lack of standardized definitions for structure-based statistical indicators, also pose challenges to the investigation of higher-order networks. In recognition of these challenges, this paper presents a comprehensive survey of commonly employed statistics and their underlying physical significance in two prevalent types of higher-order networks: hypergraphs and simplicial complex networks. This paper not only outlines the specific calculation methods and application scenarios of these statistical indicators, but also provides a glimpse into future research trends. This comprehensive overview serves as a valuable resource for beginners or cross-disciplinary researchers interested in higher-order networks, enabling them to swiftly grasp the fundamental statistics pertaining to these advanced structures. By promoting a deeper understanding of higher-order networks, this paper facilitates quantitative analysis of their structural characteristics and provides guidance for researchers who aim to develop new statistical methods for higher-order networks.
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Keywords:
- higher-order network/
- hypergraph/
- simplicial network/
- statistics
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指标类型 指标名称 度相关指标 度、超度、超边度、余平均度 聚集系数 节点的聚集系数、网络的聚集系数 距离相关指标 路径长度、超节点之间的距离 密度相关指标 超边密度、超图密度 曲率相关指标 Forman-Ricci曲率、Ollivier-Ricci曲率 中心性指标 度中心性、核心度中心性、接近中心性、
介数中心性、特征向量中心性熵相关指标 超图熵、超图的香农熵、加权超图的超图熵 
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指标类型 指标名称 度相关指标 上邻接度、下邻接度、度、上p邻接度、下p邻接度、严格上p邻接度、严格下p邻接度、
上$(h, p)$邻接度、下$(h, p)$邻接度、p邻接度、最大p邻接度、最大单纯形度路径和距离相关指标 $s_k$游走、p游走、最短路径长度、离心率、直径 中心性指标 度中心性、特征向量中心性、Katz中心性、接近中心性、介数中心性 聚集系数 聚集系数 拓扑不变量 贝蒂数、欧拉示性数 下载:
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