{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T22:15:42Z","timestamp":1767651342397,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,20]],"date-time":"2022-02-20T00:00:00Z","timestamp":1645315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004189","name":"Max Planck Society","doi-asserted-by":"publisher","award":["Max Planck Partner Group in Mathematical Biology"],"award-info":[{"award-number":["Max Planck Partner Group in Mathematical Biology"]}],"id":[{"id":"10.13039\/501100004189","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001736","name":"German-Israeli Foundation for Scientific Research and Development","doi-asserted-by":"publisher","award":["I-1514-304.6\/2019"],"award-info":[{"award-number":["I-1514-304.6\/2019"]}],"id":[{"id":"10.13039\/501100001736","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this contribution, we represent hypergraphs as partially ordered sets or posets, and provide a geometric framework based on posets to compute the Forman\u2013Ricci curvature of vertices as well as hyperedges in hypergraphs. Specifically, we first provide a canonical method to construct a two-dimensional simplicial complex associated with a hypergraph, such that the vertices of the simplicial complex represent the vertices and hyperedges of the original hypergraph. We then define the Forman\u2013Ricci curvature of the vertices and the hyperedges as the scalar curvature of the associated vertices in the simplicial complex. Remarkably, Forman\u2013Ricci curvature has a simple combinatorial expression and it can effectively capture the variation in symmetry or asymmetry over a hypergraph. Finally, we perform an empirical study involving computation and analysis of the Forman\u2013Ricci curvature of hyperedges in several real-world hypergraphs. We find that Forman\u2013Ricci curvature shows a moderate to high absolute correlation with standard hypergraph measures such as eigenvector centrality and cardinality. Our results suggest that the notion of Forman\u2013Ricci curvature extended to hypergraphs in this work can be used to gain novel insights on the organization of higher-order interactions in real-world hypernetworks.<\/jats:p>","DOI":"10.3390\/sym14020420","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T08:34:47Z","timestamp":1645432487000},"page":"420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Poset-Based Approach to Curvature of Hypergraphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Yasharth","family":"Yadav","sequence":"first","affiliation":[{"name":"The Institute of Mathematical Sciences (IMSc), Chennai 600113, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6796-9604","authenticated-orcid":false,"given":"Areejit","family":"Samal","sequence":"additional","affiliation":[{"name":"The Institute of Mathematical Sciences (IMSc), Chennai 600113, India"},{"name":"Homi Bhabha National Institute (HBNI), Mumbai 400094, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emil","family":"Saucan","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, ORT Braude College of Engineering, Karmiel 2161002, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1073\/pnas.98.2.404","article-title":"The structure of scientific collaboration networks","volume":"98","author":"Newman","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bianconi, G. (2021). Higher-Order Networks, Cambridge University Press. Elements in Structure and Dynamics of Complex Networks.","DOI":"10.1017\/9781108770996"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2020.05.004","article-title":"Networks beyond pairwise interactions: Structure and dynamics","volume":"874","author":"Battiston","year":"2020","journal-title":"Phys. Rep."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1137\/20M1355896","article-title":"The why, how, and when of representations for complex systems","volume":"63","author":"Torres","year":"2021","journal-title":"SIAM Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s00454-002-0694-2","article-title":"Isoperimetric constants of infinite plane graphs","volume":"28","author":"Lawrencenko","year":"2002","journal-title":"Discret. Comput. Geom."},{"key":"ref_8","first-page":"221","article-title":"Isoperimetric constants of (d,f)-regular planar graphs","volume":"9","author":"Higuchi","year":"2003","journal-title":"Interdiscip. Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1038\/s42254-020-00264-4","article-title":"Network geometry","volume":"3","author":"Boguna","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"41974","DOI":"10.1038\/srep41974","article-title":"Emergent hyperbolic network geometry","volume":"7","author":"Bianconi","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1140\/epjds\/s13688-017-0104-x","article-title":"Topological analysis of data","volume":"6","author":"Patania","year":"2017","journal-title":"EPJ Data Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8650","DOI":"10.1038\/s41598-018-27001-3","article-title":"Comparative analysis of two discretizations of Ricci curvature for complex networks","volume":"8","author":"Samal","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"201734","DOI":"10.1098\/rsos.201734","article-title":"Network geometry and market instability","volume":"8","author":"Samal","year":"2021","journal-title":"R. Soc. Open Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9800","DOI":"10.1038\/s41598-019-46079-x","article-title":"Ollivier-ricci curvature-based method to community detection in complex networks","volume":"9","author":"Sia","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_15","first-page":"706","article-title":"Forman\u2019s Ricci curvature\u2014From networks to hypernetworks","volume":"Volume 812","author":"Saucan","year":"2019","journal-title":"International Conference on Complex Networks and Their Applications"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12466","DOI":"10.1038\/s41598-020-68619-6","article-title":"Ollivier ricci curvature of directed hypergraphs","volume":"10","author":"Eidi","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0378-8733(91)90018-O","article-title":"Simultaneous group and individual centralities","volume":"13","author":"Bonacich","year":"1991","journal-title":"Soc. Netw."},{"key":"ref_18","unstructured":"Bloch, E. (2014). Combinatorial Ricci Curvature for Polyhedral Surfaces and Posets. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Rosen, K.H., Michaels, J.G., Gross, J.L., Grossman, J.W., and Shier, D.R. (1999). Handbook of Discrete and Combinatorial Mathematics, CRC Press.","DOI":"10.1201\/9781439832905"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Stanley, R. (1997). Enumerative Combinatorics, Cambridge University.","DOI":"10.1017\/CBO9780511805967"},{"key":"ref_21","first-page":"605","article-title":"Discrete Curvatures and Network Analysis","volume":"80","author":"Saucan","year":"2018","journal-title":"MATCH"},{"key":"ref_22","unstructured":"Hudson, J.F.P. (1969). Piecewise Linear Topology, W. A. Benjamin."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rourke, C.P., and Sanderson, B.J. (1972). Introduction to Piecewise-Linear Topology, Springer.","DOI":"10.1007\/978-3-642-81735-9"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s00454-002-0743-x","article-title":"Bochner\u2019s method for cell complexes and combinatorial Ricci curvature","volume":"29","author":"Forman","year":"2003","journal-title":"Discret. Comput. Geom."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Jost, J. (2017). Riemannian Geometry and Geometric Analysis, Springer. [7th ed.].","DOI":"10.1007\/978-3-319-61860-9"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"P063206","DOI":"10.1088\/1742-5468\/2016\/06\/063206","article-title":"Forman curvature for complex networks","volume":"2016","author":"Sreejith","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1093\/comnet\/cnx049","article-title":"Coarse geometry of evolving networks","volume":"6","author":"Weber","year":"2018","journal-title":"J. Complex Netw."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"D535","DOI":"10.1093\/nar\/gkj109","article-title":"BioGRID: A general repository for interaction datasets","volume":"34","author":"Stark","year":"2006","journal-title":"Nucleic Acids Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"D559","DOI":"10.1093\/nar\/gky973","article-title":"CORUM: The comprehensive resource of mammalian protein complexes-2019","volume":"47","author":"Giurgiu","year":"2018","journal-title":"Nucleic Acids Res."},{"key":"ref_30","first-page":"D845","article-title":"The DisGeNET knowledge platform for disease genomics: 2019 update","volume":"48","author":"Ronzano","year":"2019","journal-title":"Nucleic Acids Res."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kunegis, J. (2013, January 13\u201317). KONECT: The Koblenz Network Collection. Proceedings of the 22nd International Conference on World Wide Web (WWW \u201913 Companion), Rio de Janeiro, Brazil.","DOI":"10.1145\/2487788.2488173"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0378-8733(96)00300-0","article-title":"Centrality in affiliation networks","volume":"19","author":"Faust","year":"1997","journal-title":"Soc. Netw."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"E11221","DOI":"10.1073\/pnas.1800683115","article-title":"Simplicial closure and higher-order link prediction","volume":"115","author":"Benson","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.scaman.2010.10.002","article-title":"For the few not the many? The effects of affirmative action on presence, prominence, and social capital of women directors in Norway","volume":"27","author":"Seierstad","year":"2011","journal-title":"Scand. J. Manag."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mislove, A., Marcon, M., Gummadi, K.P., Druschel, P., and Bhattacharjee, B. (2007, January 24\u201326). Measurement and Analysis of Online Social Networks. Proceedings of the 5th ACM\/Usenix Internet Measurement Conference (IMC\u201907), San Diego, CA, USA.","DOI":"10.1145\/1298306.1298311"},{"key":"ref_36","unstructured":"Stewart, C., and Woon, J. (2022, February 10). Congressional Committee Assignments, 103rd to 114th Congresses, 1993\u20132017. Available online: http:\/\/web.mit.edu\/17.251\/www\/data_page.html."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.socnet.2011.07.001","article-title":"Triadic closure in two-mode networks: Redefining the global and local clustering coefficients","volume":"35","author":"Opsahl","year":"2013","journal-title":"Soc. Netw."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Boulicaut, J.F., Esposito, F., Giannotti, F., and Pedreschi, D. (2004). The Enron Corpus: A New Dataset for Email Classification Research. Machine Learning: ECML 2004, Springer.","DOI":"10.1007\/b100702"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Mastrandrea, R., Fournet, J., and Barrat, A. (2015). Contact Patterns in a High School: A Comparison between Data Collected Using Wearable Sensors, Contact Diaries and Friendship Surveys. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0136497"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"eabh1303","DOI":"10.1126\/sciadv.abh1303","article-title":"Generative hypergraph clustering: From blockmodels to modularity","volume":"7","author":"Chodrow","year":"2021","journal-title":"Sci. Adv."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1080\/15427951.2004.10129091","article-title":"Deeper Inside PageRank","volume":"1","author":"Langville","year":"2003","journal-title":"Internet Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1750013","DOI":"10.1142\/S0129065717500137","article-title":"From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity","volume":"28","author":"Fletcher","year":"2018","journal-title":"Int. J. Neural Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-019-50202-3","article-title":"Persistent homology of unweighted complex networks via discrete Morse theory","volume":"9","author":"Kannan","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"110260","DOI":"10.1016\/j.chaos.2020.110260","article-title":"Forman\u2013Ricci curvature and persistent homology of unweighted complex networks","volume":"140","author":"Roy","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Saucan, E. (2020). Discrete Morse Theory, Persistent Homology and Forman\u2013Ricci Curvature. arXiv.","DOI":"10.4310\/MCGD.2021.v1.n2.a1"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/420\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:23:23Z","timestamp":1760135003000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/420"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,20]]},"references-count":45,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020420"],"URL":"https:\/\/doi.org\/10.3390\/sym14020420","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,2,20]]}}}