Abstract
The concept of distance is a fundamental idea in graphs and hypergraphs. However, its extension to weighted hypergraphs is challenging, since it may result in inconsistencies, especially if the weights are arbitrarily assigned to the hyperedges. We address this challenge by proposing a general distance measure for weighted hypergraphs. Our measure is well-defined, and it reduces to the classic graph distance if the edges in the hypergraph are all pairwise links. We demonstrate the applicability of our definition by analyzing a number of real-world higher-order datasets, including the network of preprints in the arXiv repository, for which we choose the weights in a way that reflects the notion of cognitive distance, which measures the conceptual remoteness between fields. Our results demonstrate that, when higher-order edges cannot be neglected, the use of a full hypergraph measure is necessary to avoid the information loss that would result from commonly used approaches, such as clique projection.
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Data availability
The datasets used in this study are available at https://www.cs.cornell.edu/arb/data/. The arXiv dataset is available at https://github.com/mattbierbaum/arxiv-public-datasets.
Code availability
The source code for all the computations is available at https://codeberg.org/paraw/Hyperdist/.
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Acknowledgements
A.M.R. was supported by the Ministry of Science and Higher Education of the Russian Federation, project No. FSMG-2024-0011. C.I.D.G. acknowledges funding from the Bulgarian Ministry of Education and Science, under Project No. BG-RRP-2.004-0006-C02. S.B. acknowledges support from the project n. PGR01177 of the Italian Ministry of Foreign Affairs and International Cooperation. The authors would like to thank Ivan Samoylenko and Kirill Kovalenko for fruitful discussions.
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C.I.D.G.: validation, formal analysis, supervision, writing. E.V.: methodology, numerics, formal analysis, writing. L.T.: methodology, numerics, formal analysis, writing. D.F.: numerics, writing. D.M.: validation, writing, project administration. A.M.R.: validation, writing. S.B.: conceptualization, supervision, writing.
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del Genio, C.I., Vasilyeva, E., Tupikina, L. et al. Distances in weighted higher-order networks. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02592-w
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DOI: https://doi.org/10.1038/s42005-026-02592-w
