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 a doubly exponential candidate space in BBP to a linear number of singly exponential edge-subset searches in TPP, 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 h. 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.

Security engineering has shifted from the paradigm of securing a list of assets to defending interconnected graphs. In this new paradigm, security researchers have employed higher-order graphs such as directed hypergraphs, where nodes represent sets of elements, and directed edges connect these sets. Higher-order graphs are unmatched in their flexibility to naturally model a broad range of network systems where higherorder relationships exist among their interacting parts. Despite their powerful expressiveness, higher-order graphs have been underexplored in the security modeling literature because of their inherent complexity and the lack of appropriate tools and algorithms. In this paper, we give an introduction to various types of higher-order graphs, including metagraphs, directed hypergraphs, and explore their relationships with similar concepts such as Petri nets. We also present the applications of higherorder graphs in a couple of network security modeling problems with recently developed algorithms for analyzing these types of graphs.

The Waxman random graph is a generalisation of the simple Erdős-Rényi or Gilbert random graph. It is useful for modelling physical networks where the increased cost of longer links means they are less likely to be built, and thus less numerous than shorter links. The model has been in continuous use for over two decades with many attempts to select parameters which match real networks. In most the parameters have been arbitrarily selected, but there are a few cases where they have been calculated using a formal estimator. However, the performance of the estimator was not evaluated in any of these cases. This paper presents both the first evaluation of formal estimators for the parameters of these graphs, and a new Maximum Likelihood Estimator with $O(n)$ computational time complexity that requires only link lengths as input.

Spatially Embedded Random Networks such as the Waxman random graph have been used in many settings for synthesizing networks. Prior to our work, there existed no software for generating these efficiently. Existing techniques are $O(n^2)$ where $n$ is the number of nodes in the network; in this paper we present an $O(n + e)$ algorithm, where $e$ is the number of edges.

In the past few years, significant progress has been made in using software defined networking to increase automation, improve network security, simplify network configuration and reduce human effort to establish and maintain the network. There are now a vast number of studies exploring how to utilise policies to achieve the above goals. In this paper, we apply policy defined networking to wireless access network functions. We describe the details of our prototype policy defined networking solution that automatically translates high-level policies into device level implementations. We develop a novel metagraph model that can be used for policy specification, verification and refinement. We show that sophisticated traffic engineering policies can be implemented automatically on commodity hardware using our framework.

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.