This work focuses on higher-order subgraphs embedded in conventional pairwise networks, rather than on higher-order network models such as hypergraphs or simplicial complexes. Compared with conventional node- and edge-based network measures, higher-order subgraph structural characterization can capture local interaction patterns, structural roles, and mesoscopic organizational units formed by multiple nodes, thereby revealing richer organizational principles in complex networks. Motifs and graphlets are two representative types of higher-order subgraph structures and have been widely applied to network comparison, node classification, link prediction, community detection, biological network analysis, social network mining, and other tasks. However, existing studies are often scattered across different definition systems, feature indicators, and modeling methods. A systematic review of the relationship between decoupled structural characterization and coupled intelligent characterization of higher-order subgraphs is still lacking, which hinders the formation of a unified theoretical framework and methodological system.
Focusing on the characterization of higher-order subgraphs in complex networks, this work takes motifs and graphlets as representative objects and provides a systematic review from two complementary perspectives: decoupled structural characterization and coupled intelligent characterization. First, this work summarizes decoupled characterization indicators of higher-order subgraphs at different scales, including the node, edge, community, and network levels. These indicators include the frequency and orbit roles of nodes participating in motifs or graphlets, the connection functions of edges in higher-order structures, higher-order organizational patterns within and between communities, and global higher-order structural statistics at the network level. These indicators can extract higher-order topological information with explicit structural meanings from multiple scales, thereby providing fundamental feature support for complex network structure interpretation and task modeling. Second, this work further reviews coupled characterization methods for higher-order subgraph features, summarizes the technical evolution from manually designed heuristic feature combinations and traditional machine-learning-based feature fusion to deep-learning- and graph-neural-network-driven intelligent characterization models, and analyzes the characteristics of different methods in terms of feature integration ability, task adaptability, and interpretability.
Higher-order subgraph characterization provides an important bridge between local topological structures and global network functions. Decoupled characterization methods emphasize clear structural meanings and strong interpretability, and can reveal the specific roles of nodes, edges, communities, and networks in higher-order organizational patterns. In contrast, coupled intelligent characterization methods focus more on the integrated utilization of multi-scale and multi-type higher-order features, thereby improving representation learning ability and predictive performance in complex network analysis tasks. Meanwhile, this field still faces several key challenges, including the high computational complexity of higher-order subgraph extraction and statistics, the insufficient adaptability of existing methods to heterogeneous, dynamic, weighted, directed, and multilayer networks, and the limited structural interpretability of deep coupled characterization models. Future research should further develop scalable, transferable, and interpretable intelligent characterization methods for higher-order subgraphs, and promote the deep integration of higher-order structural information with artificial intelligence models. This work provides a systematic reference for higher-order structure extraction, feature fusion, task modeling, and result interpretation in complex networks, and offers insights into the construction of a unified intelligent characterization framework oriented toward multi-scale network structural semantics.