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 chain analysis, offering a clearer, more efficient framework for hypergraph simplification and analysis.

Network design, as it is currently practiced, involves putting devices together to create a network. However, a network is more than the sum of its parts, both in terms of the services it provides, and the potential for bugs. Devices are important, but their combination into a network should follow from expression of high-level policy, not the minutiae of network device configuration. Ideally we want to consider the network as a whole object. In this paper we develop generalized graph products that allow the mathematical design of a network in terms of small subgraphs that directly express business policy. The result is a flexible algebraic description of networks suitable for manipulation and proof. The approach is more than just design - it allows for analysis of existing networks providing an understanding of the policies used in their construction, something which can be difficult if the original designers no longer work on that network. We apply the approach to several real world networks to demonstrate how it can provide insight, and improve design.