The microscale organization of directed hypergraphs

TL;DR

This work provides a framework to characterize the structural organization of directed higher-order networks at their microscale, and forms reciprocity in hypergraphs, including exact, strong, and weak definitions, to measure to which extent hyperedges are reciprocated.

Abstract

Many real-world complex systems are characterized by non-pairwise -- higher-order -- interactions among system's units, and can be effectively modeled as hypergraphs. Directed hypergraphs distinguish between source and target sets within each hyperedge, and allow to account for the directional flow of information between nodes. Here, we provide a framework to characterize the structural organization of directed higher-order networks at their microscale. First, 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 formulate reciprocity in hypergraphs, including exact, strong, and weak definitions, to measure to which extent 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.

Figures (15)

• Figure 1: Schematic of a directed hypergraph. Each interaction encodes a source set of units acting towards a target set of units. We distinguish four types of directed higher-order interactions: one-to-one (black), one-to-many (blue), many-to-one (red), and many-to-many (green).
• Figure 2: Hyperedge signature of directed hypergraphs. a) We describe each system with a hyperedge signature vector whose entries encode the count of directed hyperedges with source and tail sizes $(|S|,|T|)$. We compute statistics using hyperedges with total cardinality at most $6$ (i.e., $|S|+|T|\le 6$). For visualization, we display each vector as a sequence of histogram panels: one panel for each source size $|S|=i$, separated by a small gap; within each panel, bins correspond to tail sizes $|T|$ in increasing order, restricted by $i+|T|\le 6$. Systems from the same domain share the color. b) Dendrogram resulting from agglomerative clustering applied to the correlation matrix of hyperedge signature vectors for each dataset. Correlation values are color-coded, with high positive correlations in red and high negative correlations in blue.
• Figure 3: Overlap across domains. a) Distribution of node counts within the joint z-score space of source and target overlap, representing how much nodes deviate from null model expectations in both dimensions. b) Bar plots quantifying the fraction of nodes exceeding the $\textit{z-score} \geq 2$ threshold for either source or target overlaps, nodes exceeding both thresholds, and nodes below both thresholds.
• Figure 4: Reciprocity measures for directed hypergraphs. a) Example of a directed hyperedge. b) All possible ways in which this hyperedge can be reciprocated according to our definitions. Exact reciprocity: a single hyperedge with source and target sets swapped, represented by reversing the arrow between the same node sets. Strong reciprocity: multiple hyperedges collectively reverse the interaction, possibly involving external nodes. Weak reciprocity: at least one node in the target set reciprocates with one node in the source set, illustrated as a pairwise link with reversed direction. In all panels, shaded areas group nodes involved in each interaction; colors encode the interaction pattern (green many-to-many, red many-to-one, black one-to-one); arrows encode direction from source to target. Grey disks denote nodes in the original hyperedge, and external nodes that appear only in reciprocal interactions are white with a dashed border.
• Figure 5: Higher-order reciprocity in real-world hypergraphs. a) Reciprocity score across datasets and reciprocity definitions. Each column corresponds to a distinct notion of higher-order reciprocity, thereby inducing a ranking of the datasets based on their scores. Datasets from the same domains share the same color. Arrows link the datasets across different definitions. b) Reciprocity score disaggregated by hyperedge size for each different notion of reciprocity. Trends should be interpreted relative to the null (zero line). To simplify the plots, we aggregate systems from the same domain.