Fundamental statistics of higher-order networks: a survey

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.