FORMAN–RICCI CURVATURE FOR HYPERGRAPHS

Wilmer Leal, Guillermo Restrepo, Peter F. Stadler and Jürgen Jost
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Wilmer Leal: Bioinformatics Group, Department of Computer Science, Universität Leipzig, Härtelstraße 16-18, 04107 Leipzig, Germany†Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
Guillermo Restrepo: #x2020;Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany‡Interdisciplinary Center for Bioinformatics, Universität Leipzig, Härtelstraße 16-18, 04107 Leipzig, Germany
Peter F. Stadler: Bioinformatics Group, Department of Computer Science, Universität Leipzig, Härtelstraße 16-18, 04107 Leipzig, Germany†Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany‡Interdisciplinary Center for Bioinformatics, Universität Leipzig, Härtelstraße 16-18, 04107 Leipzig, Germany§Institute for Theoretical Chemistry, University of Vienna Währingerstraße 17, 1090 Vienna, Austria¶Facultad de Ciencias, Universidad Nacional de Colombia, KR 30-45 3, 111321, Bogotá, Colombia∥The Santa Fe Institute, 1399 Hyde Park Rd, 87501, Santa Fe, New Mexico, USA
Jürgen Jost: #x2020;Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany∥The Santa Fe Institute, 1399 Hyde Park Rd, 87501, Santa Fe, New Mexico, USA

Advances in Complex Systems (ACS), 2021, vol. 24, issue 01, 1-24

Abstract: Hypergraphs serve as models of complex networks that capture more general structures than binary relations. For graphs, a wide array of statistics has been devised to gauge different aspects of their structures. Hypergraphs lack behind in this respect. The Forman–Ricci curvature is a statistics for graphs based on Riemannian geometry, which stresses the relational character of vertices in a network by focusing on the edges rather than on the vertices. Despite many successful applications of this measure to graphs, Forman–Ricci curvature has not been introduced for hypergraphs. Here, we define the Forman–Ricci curvature for directed and undirected hypergraphs such that the curvature for graphs is recovered as a special case. It quantifies the trade-off between hyperedge (arc) size and the degree of participation of hyperedge (arc) vertices in other hyperedges (arcs). Here, we determine upper and lower bounds for Forman–Ricci curvature both for hypergraphs in general and for graphs in particular. The measure is then applied to two large networks: the Wikipedia vote network and the metabolic network of the bacterium Escherichia coli. In the first case, the curvature is governed by the size of the hyperedges, while in the second example, it is dominated by the hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorylation is the metabolic bottle neck reaction but that the reverse reaction is not similarly central for the metabolism. Furthermore, we show the utility of the Forman–Ricci curvature for quantification of assortativity in hypergraphs and illustrate the idea by investigating three metabolic networks.

Keywords: Ricci curvature; hyperedge; hypergraphs; higher-order networks; hypergraph assortativity; wikipedia vote network; metabolic networks (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S021952592150003X

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