The microscale organization of directed hypergraphs

Many real-world systems exhibit higher-order interactions beyond pairwise links. Such interactions are modeled by undirected hypergraphs where edges can connect any number of vertices, but without capturing the directional nature of many real-world interactions. Directed hypergraphs overcome this limitation by distinguishing source and target sets within each hyperedge, enabling analysis of directional information flow. Here, we provide a framework to characterize the structural organization of directed higher-order networks at their microscale. We extract the fingerprint of a directed hypergraph, capturing the frequency of hyperedges with a certain source and target sizes, and use this information to compute differences in higher-order connectivity patterns among real-world systems. Then, we investigate the overlap among sources and targets to reveal recurring sets of co-sending and co-receiving nodes. We define reciprocity in hypergraphs using exact, strong, and weak definitions to quantify the extent to which hyperedges are reciprocated. Finally, we extend motif analysis to identify recurring interaction patterns and extract the building blocks of directed hypergraphs. We validate our framework on empirical datasets, including Bitcoin transactions, metabolic networks, and citation data, revealing structural principles behind the organization of real-world systems. Many real-world systems involve higher-order, directional interactions that are naturally modeled by directed hypergraphs. The authors develop a framework that characterizes directed hypergraphs through hyperedge pattern frequencies, frequent co-sending and co-receiving nodes, higher-order reciprocity, and motifs, revealing their microscale organization.