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:	Marzieh Eidi, Jürgen Jost
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2020
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/c94bf359ea57455db3718648286883bf
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spelling	oai:doaj.org-article:c94bf359ea57455db3718648286883bf2021-12-02T16:06:39ZOllivier Ricci curvature of directed hypergraphs10.1038/s41598-020-68619-62045-2322https://doaj.org/article/c94bf359ea57455db3718648286883bf2020-07-01T00:00:00Zhttps://doi.org/10.1038/s41598-020-68619-6https://doaj.org/toc/2045-2322Abstract 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.Marzieh EidiJürgen JostNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 10, Iss 1, Pp 1-14 (2020)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Marzieh Eidi Jürgen Jost Ollivier Ricci curvature of directed hypergraphs
description	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.
format	article
author	Marzieh Eidi Jürgen Jost
author_facet	Marzieh Eidi Jürgen Jost
author_sort	Marzieh Eidi
title	Ollivier Ricci curvature of directed hypergraphs
title_short	Ollivier Ricci curvature of directed hypergraphs
title_full	Ollivier Ricci curvature of directed hypergraphs
title_fullStr	Ollivier Ricci curvature of directed hypergraphs
title_full_unstemmed	Ollivier Ricci curvature of directed hypergraphs
title_sort	ollivier ricci curvature of directed hypergraphs
publisher	Nature Portfolio
publishDate	2020
url	https://doaj.org/article/c94bf359ea57455db3718648286883bf
work_keys_str_mv	AT marzieheidi ollivierriccicurvatureofdirectedhypergraphs AT jurgenjost ollivierriccicurvatureofdirectedhypergraphs
_version_	1718384930469183488

Ollivier Ricci curvature of directed hypergraphs

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