Ricci curvature of random and empirical directed hypernetworks
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Applied Network Science
RESEARCH
Open Access
Ricci curvature of random and empirical directed hypernetworks Wilmer Leal1,2
, Marzieh Eidi2
*Correspondence: [email protected] 2 Max Planck Insitute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany Full list of author information is available at the end of the article
*
and Jürgen Jost2,3
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. Keywords: Directed hypergraphs, Discrete curvature, Ricci curvature, Forman-Ricci curvature, Ollivier-Ricci curvature, Random models of directed hypergraphs, Metabolic networks
Introduction Network analysis has placed special emphasis on properties of nodes. Since networks, represented by graphs, are widely used to model discrete systems whose structure is given by relationships among objects, we shall develop tools that allow a complementary analysis of networks focused on properties of edges. Undirected graphs are only the simplest type of model for relations between discrete entities. Many empirical data expressing such relations have more structure than that of an undirected graph (Spivak 2009). For instance, the relations could be directed and/or weighted. Moreover, a relation could also involve more than two entities, as for instance in coauthorship networks or chemical reactions. Such relations can be modelled by hypergraphs rather than graphs. The hypergraph of coauthorships is undirected, whereas that of chemical reactions is directed, because a reaction may proceed from educts to © The Author(s). 2020 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or oth
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