Transitivity Preserving Projection in Directed Hypergraphs

2025-09-04

AI Summary

Directed hypergraphs suffer from severe edge explosion and loss of transitivity when modeling complex n-ary relationships—particularly at scale—hindering visualization and analysis. To address this, we propose a minimal, transitively complete projection method that preserves transitivity while enabling compact representation. Our approach extracts irreducible dominant meta-paths to achieve structural concision and introduces the first complete projection model that simultaneously retains all direct connections and the full transitive closure with a provably minimal edge set. Algorithmically, we leverage a set-trie data structure to achieve linear-time meta-path enumeration and incorporate a polynomial-time filtering mechanism for scalability. Experiments demonstrate that our method projects large real-world hypergraphs in seconds, whereas conventional approaches fail to converge within 24 hours. The resulting projections significantly improve structural fidelity, interpretability, and computational scalability—enabling practical application in domains such as cybersecurity and supply chain analysis.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Directed hypergraphs are vital for modeling complex polyadic relationships in domains such as discrete mathematics, computer science, network security, and systems modeling. However, their inherent complexity often impedes effective visualization and analysis, particularly for large graphs. This paper introduces a novel Transitivity Preserving Projection (TPP) to address the limitations of the computationally intensive Basu and Blanning projection (BBP), which can paradoxically increase complexity by flattening transitive relationships. TPP offers a minimal and complete representation of relationships within a chosen subset of elements, capturing only irreducible dominant metapaths to ensure the smallest set of edges while preserving all essential transitive and direct connections. This approach significantly enhances visualization by reducing edge proliferation and maintains the integrity of the original hypergraph's structure. We develop an efficient algorithm leveraging the set-trie data structure, reducing the computational complexity from an exponential number of metapath searches in BBP to a linear number of metapath searches with polynomial-time filtering, enabling scalability for real-world applications. Experimental results demonstrate TPP's superior performance, completing projections in seconds on graphs where BBP fails to terminate within 24 hours. By providing a minimal yet complete view of relationships, TPP supports applications in network security and supply

Problem

• Minimizing edge complexity in directed hypergraph visualization
• Preserving transitive relationships during projection efficiently
• Reducing computational costs for large-scale hypergraph analysis

Innovation

• Minimal complete representation with irreducible dominant metapaths
• Efficient algorithm using set-trie data structure
• Linear metapath searches with polynomial-time filtering