Ollivier Ricci curvature of directed hypergraphs
Abstract Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propos...
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| Autores principales: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
Nature Portfolio
2020
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/c94bf359ea57455db3718648286883bf |
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| Sumario: | Abstract Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier’s definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. While the definition looks somewhat complex, in the end we shall be able to express our curvature in a very simple formula, $$\kappa =\mu _0-\mu _2-2\mu _3$$ κ=μ0-μ2-2μ3 . This formula simply counts the fraction of vertices that have to be moved by distances 0, 2 or 3 in an optimal transport plan. We can then characterize various classes of hypergraphs by their curvature. |
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