Ricci curvature of random and empirical directed hypernetworks
Abstract Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. In this work, we systematically develop formal tools for analyzing the geometry and the dynamics of hypergraphs. In particular, we s...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-09-01
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| Series: | Applied Network Science |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/s41109-020-00309-8 |
| _version_ | 1819226781331226624 |
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| author | Wilmer Leal Marzieh Eidi Jürgen Jost |
| author_facet | Wilmer Leal Marzieh Eidi Jürgen Jost |
| author_sort | Wilmer Leal |
| collection | DOAJ |
| description | Abstract Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. 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 E. coli 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. |
| first_indexed | 2024-12-23T10:30:57Z |
| format | Article |
| id | doaj.art-2a2a4bbccf164be294966b604b1fb885 |
| institution | Directory Open Access Journal |
| issn | 2364-8228 |
| language | English |
| last_indexed | 2024-12-23T10:30:57Z |
| publishDate | 2020-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Applied Network Science |
| spelling | doaj.art-2a2a4bbccf164be294966b604b1fb8852022-12-21T17:50:25ZengSpringerOpenApplied Network Science2364-82282020-09-015111410.1007/s41109-020-00309-8Ricci curvature of random and empirical directed hypernetworksWilmer Leal0Marzieh Eidi1Jürgen Jost2Bioinformatics group, Leipzig UniversityMax Planck Insitute for Mathematics in the SciencesMax Planck Insitute for Mathematics in the SciencesAbstract Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. 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 E. coli 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.http://link.springer.com/article/10.1007/s41109-020-00309-8Directed hypergraphsDiscrete curvatureRicci curvatureForman-Ricci curvatureOllivier-Ricci curvatureRandom models of directed hypergraphs |
| spellingShingle | Wilmer Leal Marzieh Eidi Jürgen Jost Ricci curvature of random and empirical directed hypernetworks Applied Network Science Directed hypergraphs Discrete curvature Ricci curvature Forman-Ricci curvature Ollivier-Ricci curvature Random models of directed hypergraphs |
| title | Ricci curvature of random and empirical directed hypernetworks |
| title_full | Ricci curvature of random and empirical directed hypernetworks |
| title_fullStr | Ricci curvature of random and empirical directed hypernetworks |
| title_full_unstemmed | Ricci curvature of random and empirical directed hypernetworks |
| title_short | Ricci curvature of random and empirical directed hypernetworks |
| title_sort | ricci curvature of random and empirical directed hypernetworks |
| topic | Directed hypergraphs Discrete curvature Ricci curvature Forman-Ricci curvature Ollivier-Ricci curvature Random models of directed hypergraphs |
| url | http://link.springer.com/article/10.1007/s41109-020-00309-8 |
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