Abstract
The study of complex systems requires models that capture a hierarchy of higher-order interactions that move beyond the pairwise representation that simple networks provide. While mathematical frameworks exist for such higher-order systems, robust geometric tools to characterize their structure and organization remain underdeveloped. Here we show that the introduction of geometric measures for these structures is achieved by leveraging the non-commutative algebra of their matrix representations through the application of Connes’ spectral triplet formalism. Within this framework, we extend the spectral distance, a metric adapted from Connes’ formalism and previously applied to graphs, to higher-order networks, and additionally propose a definition of discrete curvature which explicitly depends on the spectral dimension. These serve as characterizing features of higher-order networks and complement known topological metrics. The formalism is demonstrated on a dataset of musical compositions, revealing their latent geometric structures.
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Acknowledgements
S.N., D.M., and M.E. are thankful for the discussions with Prof. Ali Chamseddine, who guided us to the appropriate literature, and with Dr. Ola Malaeb for the thorough review of the manuscript and the feedback she provided. D.M acknowledges Research Assistantship (RA) support for this work from the Center for Advanced Mathematical Sciences (CAMS) at the American University of Beirut.
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Najem, S., Mrad, D. & Elsayed, M. Geometric features of higher-order networks via the spectral triplet. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02651-2
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DOI: https://doi.org/10.1038/s42005-026-02651-2
