Fundamental statistics of higher-order networks: a survey

Liu Bo; Zeng Yu-Jie; Yang Rong-Mei; Lü Lin-Yuan

doi:10.7498/aps.73.20240270

Abstract

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.

Keywords:

Authors and contacts

1.
Institute of Fundamental and Frontier Studies, University of Electronic Science and Technology of China, Chengdu 610054, China
2.
School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China
3.
Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China

Corresponding author: Lü Lin-Yuan, linyuan.lv@ustc.edu.cn

Funds: Project supported by the STI 2030-Major Projects (Grant No. 2022ZD0211400), the Major Program of the National Natural Science Foundation of China (Grant No. T2293771), and the Science Fund for Distinguished Young Scholars of Sichuan Provincek, China (Grant No. 2023NSFSC1919).

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图 1 两种不同类型超图的简单示例　(a) 一个拥有11个节点的超图; (b) 一个拥有9个节点的3-均匀超图

Figure 1. A simple example of two different types of hypergraphs: (a) Simple hypergraph with 11 nodes; (b) 3-uniform hypergraph with 9 nodes.

DownLoad: Full-Size Img PowerPoint

图 2 单纯形网络相关示意图　(a) 一组时序高阶交互数据; (b) 一个11节点的单纯形网络; (c) 基于图(b)中单纯形网络的骨架网络; (d)一个11节点的团复形网络

Figure 2. Correlation diagrams of the simplicial network: (a) A set of temporal higher-order interaction data; (b) a simplicial network with 11 nodes; (c) a skeleton network based on the simplicial network in Fig. 2(b); (d) a clique complex with 11 nodes.

DownLoad: Full-Size Img PowerPoint

图 3 一个3条超边的超图和其对应的加权线图^[73]　(a) 一个3条超边的超图; (b) 图3(a)对应的加权线图

Figure 3. A hypergraph with 3 hyperedges and its corresponding weighted line graph^[73]: (a) A hypergraph with 3 hyperedges; (b) a weighted line graph corresponding to Fig. 3(a).

DownLoad: Full-Size Img PowerPoint

图 4 不同情形下两个超节点之间的路径示意图^[73]

Figure 4. Diagram of the path between two hypernodes in different cases^[73].

DownLoad: Full-Size Img PowerPoint

图 5 一个具有5条超边的超图的k-核分解示意图^[61]

Figure 5. Diagram of a k-core decomposition of a hypergraph with 5 hyperedges^[61].

DownLoad: Full-Size Img PowerPoint

表 1 基于超图的统计指标总结

Table 1. Summary of statistical indicators of the hypergraph

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

DownLoad: CSV

表 2 基于单纯形网络的统计指标总结

Table 2. Summary of statistical indicators of the simplicial network

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

DownLoad: CSV

map

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