{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:28:01Z","timestamp":1778495281067,"version":"3.51.4"},"reference-count":41,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"DOI":"10.13039\/501100001655","name":"Deutscher Akademischer Austauschdienst","doi-asserted-by":"publisher","award":["PhD reserarch grant application 57299294"],"award-info":[{"award-number":["PhD reserarch grant application 57299294"]}],"id":[{"id":"10.13039\/501100001655","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advs. Complex Syst."],"published-print":{"date-parts":[[2021,2]]},"abstract":"<jats:p>Hypergraphs serve as models of complex networks that capture more general structures than binary relations. For graphs, a wide array of statistics has been devised to gauge different aspects of their structures. Hypergraphs lack behind in this respect. The Forman\u2013Ricci curvature is a statistics for graphs based on Riemannian geometry, which stresses the relational character of vertices in a network by focusing on the edges rather than on the vertices. Despite many successful applications of this measure to graphs, Forman\u2013Ricci curvature has not been introduced for hypergraphs. Here, we define the Forman\u2013Ricci curvature for directed and undirected hypergraphs such that the curvature for graphs is recovered as a special case. It quantifies the trade-off between hyperedge (arc) size and the degree of participation of hyperedge (arc) vertices in other hyperedges (arcs). Here, we determine upper and lower bounds for Forman\u2013Ricci curvature both for hypergraphs in general and for graphs in particular. The measure is then applied to two large networks: the Wikipedia vote network and the metabolic network of the bacterium Escherichia coli. In the first case, the curvature is governed by the size of the hyperedges, while in the second example, it is dominated by the hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorylation is the metabolic bottle neck reaction but that the reverse reaction is not similarly central for the metabolism. Furthermore, we show the utility of the Forman\u2013Ricci curvature for quantification of assortativity in hypergraphs and illustrate the idea by investigating three metabolic networks.<\/jats:p>","DOI":"10.1142\/s021952592150003x","type":"journal-article","created":{"date-parts":[[2021,8,23]],"date-time":"2021-08-23T07:54:22Z","timestamp":1629705262000},"source":"Crossref","is-referenced-by-count":20,"title":["FORMAN\u2013RICCI CURVATURE FOR HYPERGRAPHS"],"prefix":"10.1142","volume":"24","author":[{"given":"WILMER","family":"LEAL","sequence":"first","affiliation":[{"name":"Bioinformatics Group, Department of Computer Science, Universit\u00e4t Leipzig, H\u00e4rtelstra\u00dfe 16-18, 04107 Leipzig, Germany"},{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GUILLERMO","family":"RESTREPO","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"},{"name":"Interdisciplinary Center for Bioinformatics, Universit\u00e4t Leipzig, H\u00e4rtelstra\u00dfe 16-18, 04107 Leipzig, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5016-5191","authenticated-orcid":false,"given":"PETER F.","family":"STADLER","sequence":"additional","affiliation":[{"name":"Bioinformatics Group, Department of Computer Science, Universit\u00e4t Leipzig, H\u00e4rtelstra\u00dfe 16-18, 04107 Leipzig, Germany"},{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"},{"name":"Interdisciplinary Center for Bioinformatics, Universit\u00e4t Leipzig, H\u00e4rtelstra\u00dfe 16-18, 04107 Leipzig, Germany"},{"name":"Institute for Theoretical Chemistry, University of Vienna W\u00e4hringerstra\u00dfe 17, 1090 Vienna, Austria"},{"name":"Facultad de Ciencias, Universidad Nacional de Colombia, KR 30-45 3, 111321, Bogot\u00e1, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00dcRGEN","family":"JOST","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"},{"name":"The Santa Fe Institute, 1399 Hyde Park Rd, 87501, Santa Fe, New Mexico, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,9,30]]},"reference":[{"key":"S021952592150003XBIB001","doi-asserted-by":"publisher","DOI":"10.1186\/1759-2208-3-1"},{"key":"S021952592150003XBIB002","doi-asserted-by":"publisher","DOI":"10.1126\/science.286.5439.509"},{"key":"S021952592150003XBIB003","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.98.032409"},{"key":"S021952592150003XBIB004","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-58002-9_1"},{"key":"S021952592150003XBIB005","doi-asserted-by":"publisher","DOI":"10.2174\/157340911795677611"},{"key":"S021952592150003XBIB006","doi-asserted-by":"publisher","DOI":"10.1038\/s42254-020-00264-4"},{"key":"S021952592150003XBIB007","doi-asserted-by":"publisher","DOI":"10.1039\/c3cs35505e"},{"key":"S021952592150003XBIB008","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-00080-0"},{"key":"S021952592150003XBIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s12064-020-00328-0"},{"key":"S021952592150003XBIB010","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-020-68619-6"},{"key":"S021952592150003XBIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2005.12.002"},{"key":"S021952592150003XBIB012","doi-asserted-by":"publisher","DOI":"10.1186\/s13321-018-0273-z"},{"key":"S021952592150003XBIB013","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-020-78336-9"},{"key":"S021952592150003XBIB014","doi-asserted-by":"publisher","DOI":"10.2174\/9781681080529115020017"},{"key":"S021952592150003XBIB015","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-002-0743-x"},{"key":"S021952592150003XBIB016","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(93)90045-P"},{"key":"S021952592150003XBIB017","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.10004"},{"key":"S021952592150003XBIB018","doi-asserted-by":"publisher","DOI":"10.1186\/1752-0509-1-26"},{"key":"S021952592150003XBIB019","doi-asserted-by":"publisher","DOI":"10.1038\/35036627"},{"key":"S021952592150003XBIB020","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1000385"},{"key":"S021952592150003XBIB021","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-019-0459-y"},{"key":"S021952592150003XBIB022","doi-asserted-by":"publisher","DOI":"10.1007\/s41109-020-00309-8"},{"key":"S021952592150003XBIB023","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2018.0581"},{"key":"S021952592150003XBIB026","volume-title":"Proc. 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