Curvature of Hypergraphs via Multi-Marginal Optimal Transport

Asoodeh, Shahab; Gao, Tingran; Evans, James A.

doi:10.1109/cdc.2018.8619706

Cited by 16 publications

(13 citation statements)

References 39 publications

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“…Before concluding this section, let us note that the Ollivier-Ricci curvature has no immediate generalization to hypernetworks [51,52]. However, one proposition to compute the Ollivier-Ricci curvature in hypernetworks is as follows.…”

Section: Extension Of Ollivier-ricci Curvature To Directed Networkmentioning

confidence: 99%

Discrete Ricci curvatures for directed networks

Saucan

Sreejith

Vivek-Ananth

et al. 2019

Chaos, Solitons & Fractals

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A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman-Ricci curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman-Ricci curvature of a directed edge elegantly incorporates vertex weights, edge weights and edge direction. Furthermore we have compared the Forman-Ricci curvature with the adaptation to directed networks of another discrete notion of Ricci curvature, namely, the well established Ollivier-Ricci curvature. However, the two above-mentioned curvature measures do not account for higher-order correlations between vertices. To this end, we adjusted Forman's original definition of Ricci curvature to account for directed simplicial complexes and also explored the potential of this new, augmented type of Forman-Ricci curvature, in directed complex networks. * asamal@imsc.res.in arXiv:1809.07698v1 [math.MG] 15 Sep 2018Curvature represents a central concept in geometry, and it certainly is the focal point of differential and Riemannian geometry, which formally quantifies our intuition of the deviation of an object from being flat. In the classical context, where the notion of curvature originated, several types of curvature have been considered [26]. Among the different notions of curvature, it appears that the concept of Ricci curvature is the most useful, after a proper discretization process, for the analysis of graphs or networks [14-18, 20-22, 24]. To this end, let us note that, in differential geometry, Ricci curvature first appears in the context of the so called Jacobi equation, that represents a linearization of the equation for geodesics [26] and, as such, governs both the growth of volumes and the dispersion rate of geodesics [26,27]. Furthermore, Ricci curvature is also an essential ingredient in the so called Bochner-Weitzenböck formula, that connects between the Laplacian on a manifold and its geometry, as expressed by its curvature. By its very definition through the above mentioned equation of geodesics, Ricci curvature depends on a direction, which then for graphs or networks translates into the fact that it should be a measure associated to an edge rather than to a vertex. Among the several curvature measures [13][14][15][16][17][18][19][20][21][22][28][29][30][31][32] that have been proposed for geometrical characterization of complex networks, two different discretizations of the Ricci curvature, Ollivier-...

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Section: Extension Of Ollivier-ricci Curvature To Directed Networkmentioning

confidence: 99%

Discrete Ricci curvatures for directed networks

Saucan

Sreejith

Vivek-Ananth

et al. 2019

Chaos, Solitons & Fractals

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“…Unfortunately, generalizing the notion of optimal transport from node to node along edges to higher dimensional faces is far from straightforward. Recently, a new type of curvature, the coarse scalar curvature, was introduced [1] as a first extension of Ollivier's methods to hypernetworks. We show that, based on the parametrization of hypernetworks as polyhedral complexes, it is possible to extend Ollivier's curvature to hypernetworks in a more general setting.…”

Section: Ollivier's Ricci Curvaturementioning

confidence: 99%

Forman’s Ricci Curvature - From Networks to Hypernetworks

Saucan

Weber

2018

Studies in Computational Intelligence

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Networks and their higher order generalizations, such as hypernetworks or multiplex networks are ever more popular models in the applied sciences. However, methods developed for the study of their structural properties go little beyond the common name and the heavy reliance of combinatorial tools. We show that, in fact, a geometric unifying approach is possible, by viewing them as polyhedral complexes endowed with a simple, yet, the powerful notion of curvature -the Forman Ricci curvature. We systematically explore some aspects related to the modeling of weighted and directed hypernetworks and present expressive and natural choices involved in their definitions. A benefit of this approach is a simple method of structure-preserving embedding of hypernetworks in Euclidean N-space. Furthermore, we introduce a simple and efficient manner of computing the well established Ollivier-Ricci curvature of a hypernetwork.

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“…Recently, notions inspired by Riemannian geometry appeared very promising in this direction. More precisely, it has been discovered that concepts of curvature can be formulated in such a way that they apply naturally not only to smooth Riemannian manifolds, but also to various kinds of discrete spaces (Forman 2003;Saucan 2019), like graphs (Jost and Liu 2014;Ollivier 2007) or hypergraphs (Asoodeh et al 2018;Banerjee 2020). Much effort has focused on concepts of Ricci curvature in this context, and that is also what we shall explore in this paper, drawing on recent theoretical work from our group, like notions of such Ricci curvature for directed hypergraphs (Eidi and Jost 2020;Leal et al 2018).…”

Section: Introductionmentioning

confidence: 99%

Ricci curvature of random and empirical directed hypernetworks

Leal

Eidi²,

Jost

2020

Appl Netw Sci

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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.

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Curvature of Hypergraphs via Multi-Marginal Optimal Transport

Cited by 16 publications

References 39 publications

Discrete Ricci curvatures for directed networks

Discrete Ricci curvatures for directed networks

Forman’s Ricci Curvature - From Networks to Hypernetworks

Ricci curvature of random and empirical directed hypernetworks

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