Can hypergraphs provide a new perspective on complex systems? This paper revisits the definition of differential operators on hypergraphs, extending traditional graph analysis to systems with interactions beyond simple pairs. It focuses on defining Laplacian and p-Laplace operators for both oriented and unoriented hypergraphs, exploring their fundamental characteristics, variational structure, and scale spaces. The authors demonstrate that diffusion equations on hypergraphs serve as viable models for diverse applications, such as information dissemination on social networks and image processing. Furthermore, the spectral analysis and scale spaces derived from these operators offer a promising methodology for analyzing complex data and their multiscale structure. Highlighting the need for spectral analysis and appropriate scale spaces on hypergraphs, the paper introduces a novel axiomatic approach for defining differential operators with trivial first eigenfunctions, leading to more interpretable second eigenfunctions. This property, often lacking in existing hypergraph p-Laplacians, enhances the potential for deeper insights into complex datasets.
This paper, published in the Journal of Mathematical Imaging and Vision, fits squarely within the journal's scope by advancing the mathematical foundations and applications of image processing and computer vision. The exploration of hypergraph p-Laplacians and scale spaces provides new tools for analyzing complex data, aligning with the journal's emphasis on cutting-edge techniques in mathematical imaging.