The microscale organization of directed hypergraphs

TL;DR

This paper introduces a framework for analyzing the microscale structure of directed hypergraphs, capturing higher-order interactions, reciprocity, and motifs, and applies it to real-world data like Bitcoin, metabolic, and citation networks.

Contribution

It provides novel methods to characterize higher-order connectivity, reciprocity, and motifs in directed hypergraphs, advancing understanding of complex systems' structural organization.

Findings

Identified distinct higher-order connectivity patterns in real-world networks.

Quantified reciprocity levels in directed hypergraphs across datasets.

Revealed recurring motifs that serve as building blocks of complex systems.

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.