The microscale organization of directed hypergraphs (2410.16258v1)

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.

• The paper presents a novel framework for directed hypergraphs by introducing hyperedge signature vectors to encode source-target interaction patterns.
• It redefines reciprocity with exact, strong, and weak measures, extending traditional network analysis to accommodate higher-order interactions.
• Motif analysis across diverse datasets reveals distinct structural patterns that illuminate complex feedback mechanisms in real-world systems.

The Microscale Organization of Directed Hypergraphs

This paper presents a formal framework aimed at characterizing the microscale structure of directed hypergraphs, a refined model suited for real-world systems where complex higher-order interactions occur. Traditional graph models are limited by their focus on pairwise interactions, often underrepresenting complex systems where interactions involve multiple entities. Hypergraphs, with their capacity to include multiple nodes in hyperedges, are a natural extension. The authors explore directed hypergraphs, allowing for differentiation between source and target sets, enabling the representation of directional flows in systems such as financial transactions and biological networks.

Framework Overview

The paper proposes a methodological framework to dissect the structural properties of directed hypergraphs. The following key components form the core of this exploration:

1. Hyperedge Signature Vectors: A central aspect of the paper is the creation of hyperedge signature vectors, which provide a descriptive fingerprint of a hypergraph. These vectors encode the distribution and frequency of hyperedges based on the sizes of their source and target sets. This approach allows for the differentiation of various real-world systems by their unique patterns of interaction.
2. Higher-Order Reciprocity: Extending the notion of reciprocity, the authors redefine it for hypergraphs, introducing three distinct types: exact, strong, and weak reciprocity. Each type offers a progressively relaxed criterion for considering interactions reciprocated, ranging from strictly bidirectional to loosely defined mutuality.
3. Motif Analysis: The paper advances motif analysis, traditionally limited to undirected and pairwise networks, to encompass the complexities of directed hypergraphs. This allows for the identification of recurring interaction patterns or 'motifs,' providing insight into the fundamental substructures that constitute these networks.

Key Results and Insights

• Empirical Observations: The framework is validated against datasets from diverse domains, including Bitcoin transactions, email networks, and citation networks. Each domain exhibits distinct hyperedge patterns, with motifs and reciprocity reflecting the underlying systems’ functional and structural nuances.
• Correlation of Hypergraph Structures: By comparing hyperedge signature vectors, systems can be clustered according to their relational patterns. This not only highlights domain-specific characteristics but also reveals cross-domain structural similarities.
• Reciprocity Patterns: Notably, weak reciprocity is prevalent across all systems, whereas exact reciprocity is comparatively rare. These findings suggest complex feedback mechanisms between source and target entities in systems with directed group interactions.
• Motif Abundancy: The paper of motifs reveals that systems tend to favor specific patterns of interaction, with variations between domains attributed to their unique structural and functional requirements.

Implications and Future Directions

The implications of this work are extensive. By accurately modeling complex interactions in fields such as biology and finance, directed hypergraphs can yield more precise insights into systems behavior. This capability is invaluable in scenarios like detecting fraudulent financial interactions, where the relationship dynamics are multifaceted.

Future research may expand on scalability concerns and evolve to include weighted or temporal dynamics in hypergraphs. The combination of novel sampling methods and advanced computational techniques could also facilitate the analysis of larger and more complex datasets or motif sizes beyond the current computational limits.

This paper lays critical groundwork for leveraging directed hypergraphs in the analysis of complex systems, providing a nuanced perspective towards understanding higher-order connectivity and interaction patterns in real-world networks.