Complex systems can be more accurately described by higher-order interactions among multiple units. Hypergraphs excel at depicting these interactions, surpassing the binary limitations of traditional graphs. However, retrieving valuable information from hypergraphs is often challenging due to their intricate interconnections. To address this issue, we introduce a new category of structural patterns, hypermotifs, which are defined as statistically significant local structures formed by interconnected hyperedges. We propose a systematic framework for hypermotif extraction. This framework features the encoding, census, and evaluation of higher-order patterns, effectively overcoming their inherent complexity and diversity. Our experimental results demonstrate that hypermotifs can serve as higher-order fingerprints of real-world hypergraphs, helping to identify hypergraph classes based on network functions. These motifs potentially represent preferential attachments and key modules in real-world hypergraphs, arising from specific mechanisms or constraints. Our work validates the efficacy of hypermotifs in exploring hypergraphs, offering a powerful tool for revealing the design principles and underlying dynamics of interacting systems.

Research on reservoir-unit division in fault-controlled oil and gas reservoirs is essential for analyzing reservoir hydrocarbon migration and accumulation. Currently, most research on reservoir-unit division has focused solely on the identification of faults and caves, employing three-dimensional spatial visualization or other methods for a simple analysis of their links. However, these approaches often lack a reasoning process that exploits the links between faults and caves for deeper insights. For such complex oil and gas reservoirs, a systematic analysis based on the interrelations between multiple geological factors is needed. Therefore, this paper proposes a graph-based method for reservoir-unit division in fault-controlled oil and gas reservoirs, enabling the representation of links between faults and caves, and it presents further systematic analysis to derive the reservoir-unit division results. A multi-attribute graph-clustering-based fault-extraction method is utilized to achieve comprehensive fault representations as fault entities. More reliable cave-instance segmentation results are obtained through attribute fusion, representing cavity entities. A graph incorporating fault and cave entities is then created. Fault entities are classified into several levels according to their spatial scale, and directed edges are utilized to represent connectivity links between faults and caves. Moreover, a connectivity analysis centered on caves was conducted using the created graph. Based on existing reservoir-unit knowledge and the cave-connectivity analysis results, reservoir-unit division was achieved. The proposed method provided reservoir-unit division results highly consistent with the information contained in seismic data, offering a new perspective for multielement integrated analysis in geophysical exploration.