{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T13:14:06Z","timestamp":1775913246882,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T00:00:00Z","timestamp":1686528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Rey Juan Carlos University","award":["M1967"],"award-info":[{"award-number":["M1967"]}]},{"name":"Rey Juan Carlos University","award":["M2978"],"award-info":[{"award-number":["M2978"]}]},{"name":"Rey Juan Carlos University","award":["M3033"],"award-info":[{"award-number":["M3033"]}]},{"name":"Rey Juan Carlos University","award":["NSh-775.2022.1.1"],"award-info":[{"award-number":["NSh-775.2022.1.1"]}]},{"name":"Rey Juan Carlos University","award":["ZA23GR02"],"award-info":[{"award-number":["ZA23GR02"]}]},{"name":"Rey Juan Carlos University","award":["VJR\/2019\/000034"],"award-info":[{"award-number":["VJR\/2019\/000034"]}]},{"name":"the program \u201cLeading Scientific Schools\u201d","award":["M1967"],"award-info":[{"award-number":["M1967"]}]},{"name":"the program \u201cLeading Scientific Schools\u201d","award":["M2978"],"award-info":[{"award-number":["M2978"]}]},{"name":"the program \u201cLeading Scientific Schools\u201d","award":["M3033"],"award-info":[{"award-number":["M3033"]}]},{"name":"the program \u201cLeading Scientific Schools\u201d","award":["NSh-775.2022.1.1"],"award-info":[{"award-number":["NSh-775.2022.1.1"]}]},{"name":"the program \u201cLeading Scientific Schools\u201d","award":["ZA23GR02"],"award-info":[{"award-number":["ZA23GR02"]}]},{"name":"the program \u201cLeading Scientific Schools\u201d","award":["VJR\/2019\/000034"],"award-info":[{"award-number":["VJR\/2019\/000034"]}]},{"name":"Italian Ministry of Foreign Affairs and International Cooperation","award":["M1967"],"award-info":[{"award-number":["M1967"]}]},{"name":"Italian Ministry of Foreign Affairs and International Cooperation","award":["M2978"],"award-info":[{"award-number":["M2978"]}]},{"name":"Italian Ministry of Foreign Affairs and International Cooperation","award":["M3033"],"award-info":[{"award-number":["M3033"]}]},{"name":"Italian Ministry of Foreign Affairs and International Cooperation","award":["NSh-775.2022.1.1"],"award-info":[{"award-number":["NSh-775.2022.1.1"]}]},{"name":"Italian Ministry of Foreign Affairs and International Cooperation","award":["ZA23GR02"],"award-info":[{"award-number":["ZA23GR02"]}]},{"name":"Italian Ministry of Foreign Affairs and International Cooperation","award":["VJR\/2019\/000034"],"award-info":[{"award-number":["VJR\/2019\/000034"]}]},{"name":"Vajra project","award":["M1967"],"award-info":[{"award-number":["M1967"]}]},{"name":"Vajra project","award":["M2978"],"award-info":[{"award-number":["M2978"]}]},{"name":"Vajra project","award":["M3033"],"award-info":[{"award-number":["M3033"]}]},{"name":"Vajra project","award":["NSh-775.2022.1.1"],"award-info":[{"award-number":["NSh-775.2022.1.1"]}]},{"name":"Vajra project","award":["ZA23GR02"],"award-info":[{"award-number":["ZA23GR02"]}]},{"name":"Vajra project","award":["VJR\/2019\/000034"],"award-info":[{"award-number":["VJR\/2019\/000034"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We explore the metric structure of networks with higher-order interactions and introduce a novel definition of distance for hypergraphs that extends the classic methods reported in the literature. The new metric incorporates two critical factors: (1) the inter-node distance within each hyperedge, and (2) the distance between hyperedges in the network. As such, it involves the computation of distances in a weighted line graph of the hypergraph. The approach is illustrated with several ad hoc synthetic hypergraphs, where the structural information unveiled by the novel metric is highlighted. Moreover, the method\u2019s performance and effectiveness are shown through computations on large real-world hypergraphs, which indeed reveal new insights into the structural features of networks beyond pairwise interactions. Namely, using the new distance measure, we generalize the definitions of efficiency, closeness and betweenness centrality for the case of hypergraphs. Comparing the values of these generalized measures with their analogs calculated for the hypergraph clique projections, we show that our measures provide significantly different assessments on the characteristics (and roles) of the nodes from the information-transferability point of view. The difference is brighter for hypergraphs in which hyperedges of large sizes are frequent, and nodes relating to these hyperedges are rarely connected by other hyperedges of smaller sizes.<\/jats:p>","DOI":"10.3390\/e25060923","type":"journal-article","created":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T02:56:34Z","timestamp":1686624994000},"page":"923","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Distances in Higher-Order Networks and the Metric Structure of Hypergraphs"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9303-8424","authenticated-orcid":false,"given":"Ekaterina","family":"Vasilyeva","sequence":"first","affiliation":[{"name":"The Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Institutskiy per., 9, 141701 Dolgoprudny, Moscow Region, Russia"},{"name":"P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninsky Prosp., 53, 119991 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9259-9716","authenticated-orcid":false,"given":"Miguel","family":"Romance","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, Ciencia e Ingenier\u00eda de los Materiales y Tecnolog\u00eda Electr\u00f3nica, Universidad Rey Juan Carlos, Calle Tulip\u00e1n s\/n, M\u00f3stoles, 28933 Madrid, Spain"},{"name":"Mathematical Computation Laboratory on Complex Networks and Its Appliactions, Universidad Rey Juan Carlos, Calle Tulip\u00e1n s\/n, M\u00f3stoles, 28933 Madrid, Spain"}]},{"given":"Ivan","family":"Samoylenko","sequence":"additional","affiliation":[{"name":"The Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Institutskiy per., 9, 141701 Dolgoprudny, Moscow Region, Russia"},{"name":"Faculty of Mathematics, National Research University Higher School of Economics, Usacheva str., 6, 119048 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0029-4016","authenticated-orcid":false,"given":"Kirill","family":"Kovalenko","sequence":"additional","affiliation":[{"name":"Scuola Superiore Meridionale, Largo S. Marcellino, 10, 80138 Napoli, NA, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1779-2513","authenticated-orcid":false,"given":"Daniil","family":"Musatov","sequence":"additional","affiliation":[{"name":"The Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Institutskiy per., 9, 141701 Dolgoprudny, Moscow Region, Russia"},{"name":"Institute of Economics, Mathematics and Information Technology, Russian Academy of National Economy and Public Administration, pr. Vernadskogo, 84, 119606 Moscow, Russia"},{"name":"Caucasus Mathematical Center, Adyghe State University, ul. Pervomaiskaya, 208, 385000 Maykop, The Republic of Adygea, Russia"}]},{"given":"Andrey Mihailovich","family":"Raigorodskii","sequence":"additional","affiliation":[{"name":"The Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Institutskiy per., 9, 141701 Dolgoprudny, Moscow Region, Russia"},{"name":"Caucasus Mathematical Center, Adyghe State University, ul. Pervomaiskaya, 208, 385000 Maykop, The Republic of Adygea, Russia"},{"name":"Mechanics and Mathematics Faculty, Moscow State University, Leninskie Gory, 1, 119991 Moscow, Russia"},{"name":"Institute of Mathematics and Computer Science, Buryat State University, ul. Ranzhurova, 5, 670000 Ulan-Ude, The Republic of Buryatia, Russia"}]},{"given":"Stefano","family":"Boccaletti","sequence":"additional","affiliation":[{"name":"The Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Institutskiy per., 9, 141701 Dolgoprudny, Moscow Region, Russia"},{"name":"Departamento de Matem\u00e1tica Aplicada, Ciencia e Ingenier\u00eda de los Materiales y Tecnolog\u00eda Electr\u00f3nica, Universidad Rey Juan Carlos, Calle Tulip\u00e1n s\/n, M\u00f3stoles, 28933 Madrid, Spain"},{"name":"CNR\u2014Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, FI, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. 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