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In this work, we systematically develop formal tools for analyzing the geometry and the dynamics of hypergraphs. In particular, we show that Ricci curvature concepts, inspired by the corresponding notions of Forman and Ollivier for graphs, are powerful tools for probing the local geometry of hypergraphs. In fact, these two curvature concepts complement each other in the identification of specific connectivity motifs. In order to have a baseline model with which we can compare empirical data, we introduce a random model to generate directed hypergraphs and study properties such as degree of nodes and edge curvature, using numerical simulations. We can then see how our notions of curvature can be used to identify connectivity patterns in the metabolic network of <jats:italic>E. coli<\/jats:italic> that clearly deviate from those of our random model. Specifically, by applying hypergraph shuffling to this metabolic network we show that the changes in the wiring of a hypergraph can be detected by Forman Ricci and Ollivier Ricci curvatures.<\/jats:p>","DOI":"10.1007\/s41109-020-00309-8","type":"journal-article","created":{"date-parts":[[2020,9,15]],"date-time":"2020-09-15T12:02:39Z","timestamp":1600171359000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Ricci curvature of random and empirical directed hypernetworks"],"prefix":"10.1007","volume":"5","author":[{"given":"Wilmer","family":"Leal","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marzieh","family":"Eidi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00fcrgen","family":"Jost","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,9,15]]},"reference":[{"key":"309_CR1","doi-asserted-by":"publisher","first-page":"1180","DOI":"10.1109\/CDC.2018.8619706","volume-title":"2018 IEEE Conference on Decision and Control (CDC)","author":"S Asoodeh","year":"2018","unstructured":"Asoodeh, S, Gao T, Evans J (2018) Curvature of hypergraphs via multi-marginal optimal transport In: 2018 IEEE Conference on Decision and Control (CDC), 1180\u20131185.. IEEE, Piscataway. https:\/\/doi.org\/10.1109\/CDC.2018.8619706."},{"key":"309_CR2","doi-asserted-by":"publisher","unstructured":"Banerjee, A (2020) On the spectrum of hypergraphs. Linear Algebra Appl. In Press. https:\/\/doi.org\/10.1016\/j.laa.2020.01.012.","DOI":"10.1016\/j.laa.2020.01.012"},{"key":"309_CR3","doi-asserted-by":"publisher","unstructured":"Eidi, M, Jost J (2020) Ollivier ricci curvature of directed hypergraphs. Sci Rep 10:12466. https:\/\/doi.org\/10.1038\/s41598-020-68619-6 http:\/\/arxiv.org\/abs\/1907.04727.","DOI":"10.1038\/s41598-020-68619-6"},{"key":"309_CR4","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s00454-002-0743-x","volume":"29","author":"R Forman","year":"2003","unstructured":"Forman, R (2003) Bochner\u2019s method for cell complexes and combinatorial ricci curvature. 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