Collaborative Research: CIF: Core: From Directed Hypergraph Signal Processing and Networks to Directed Acyclic Hypergraph Learning

Many of today’s most important scientific, engineering, and societal systems depend on interactions among groups, not just pairs. In biological systems, several molecules may jointly trigger another process. In logistics, multiple suppliers and facilities may jointly determine the flow of products. In communication, computing, and decision systems, groups of agents may influence other groups in ways that are directional, structured, and time ordered. Existing network models often simplify these relationships into pairwise links, which can hide the true group behavior and obscure the direction of influence. This project develops new mathematical and computational tools for modeling such directed group interactions. The resulting methods will make it possible to better analyze, predict, and interpret complex connected data in areas such as biological signaling, medical reasoning, supply-chain resilience, scheduling, communication systems, and trustworthy artificial intelligence (AI). The project will also contribute to education and workforce development by training graduate and undergraduate researchers, releasing open-source software, creating example datasets and reproducibility materials, and increasing access to instructional resources in network science, signal processing, and machine learning. The project will develop a unified framework for signal processing, structure learning, and neural network modeling on directed hypergraphs, which are network models whose links can connect a set of source nodes to a set of target nodes. The research has four main components. First, the team of researchers will define directed hypergraph operators that preserve source and target roles, prevent unintended mixing among target nodes, and support localized signal propagation. Second, the project will develop stable spectral methods for directed hypergraphs using t-product tensor algebra and self-adjoint embeddings, enabling Fourier-type transforms with energy preservation and exact reconstruction for tasks such as filtering, denoising, sampling, interpolation, and compression. Third, the project will create methods for learning directed and directed acyclic hypergraph structure from data by extending structural equation modeling to higher-order interactions, developing differentiable acyclicity constraints, and incorporating latent-variable models to improve robustness when hidden factors are present. Fourth, the project will design scalable neural network architectures for directed hypergraphs, including dual-stream convolutional and message-passing layers that maintain distinct source and target representations while avoiding explicit construction of very large tensors. The resulting theory, algorithms, software, and benchmarks are expected to advance the foundations of higher-order network analysis and provide practical tools for modeling directed, multi-entity interactions in complex scientific and engineering systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.