Fundamental statistics of higher-order networks: a survey

Complex networks serve as indispensable instruments for characterizing and comprehending 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 analyzing traditional pairwise networks from a higher-dimensional perspective. This endeavor has unearthed novel phenomena distinct from those observed in conventional pairwise networks. However, despite the significance of higher-order networks, research in this area remains comparatively sparse. Furthermore, the intricacy of higher-order interactions has led to a dearth of standardized definitions for their structural statistical measures, posing additional challenges in their investigation. 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. It 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 fostering a deeper understanding of higher-order networks, this paper facilitates quantitative analysis of their structural characteristics and serves as a guide for researchers aiming to develop novel statistical methods tailored to higher-order networks.