Modelling Methods，Applications and Perspectives on Higher-order Relationships

In recent years，big data technologies have provided crucial data support for network science，advancing the development of modeling and analytical methods for higher-order relationships in complex systems，thereby offering new avenues for revealing the underlying mechanisms of complex systems and demonstrating significant application potential. Focusing on the relationships between system elements，this paper begins with the fundamental concepts of higher-order relations by introducing the definitions of higher-order interaction relations and higher-order dependency relations，and then provides a comprehensive summary and description of modeling methods based on simplicial complexes，hypergraphs，motifs and higher-order Markov processes. Furthermore，by examining cutting-edge applications across fields such as social science，biology，and engineering physics，the current status of higher-order relation mining in real complex system studies is demonstrated. Finally，the challenges confronting higher-order relationships research are analyzed from the dual perspectives of modeling methodologies and practical applications，and future research directions are subsequently proposed.