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 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, κ = µ 0 − µ 2 − 2µ 3. 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. Principles of network analysis. Network analysis constitutes one of the success stories in the study of complex systems 3,6,11. For the mathematical analysis, a network is modelled as a (perhaps weighted and/or directed) graph. One can then look at certain graph theoretical properties of an empirical network, like its degree or motiv distribution, its assortativity or clustering coefficient, the spectrum of its Laplacian, and so on. One can also compare an empirical network with certain deterministic or random theoretical models. Successful as this analysis clearly is, we nevertheless see two important limitations. One is that many of the prominent concepts and quantities used in the analysis of empirical networks are node based, like the degree sequence. The structure of a network is encoded, however, not in its vertices or nodes, but rather in its edges, that is, the relations between the nodes. Therefore, here and in other works of our group, we wish to advocate and pursue an analysis whose fundamental ingredients are edge based quantities. Secondly, many real data sets are naturally modelled by structures that are somewhat more general than graphs, because they may contain relations involving more than two elements. For instance, chemical reactions typically involve more than two substances. This leads to hypergraphs, a subject that is currently gaining much momentum, see for instance 4. In an undirected graph, an edge is given by an unordered pair of vertices, that is, by a two-element subset of the set of all vertices. For an undirected hypergraph, a hyperedge is given by any non-empty subset of the vertex set. In a directed graph, an edge is an ordered pair of vertices, that is, it connects two vertices, its tail and its head. Analogously, in a directed hypergraph, a hyperedge connects two non-empty sets of vertices, a tail set and a head set. Directed graphs thus are special directed hypergraphs, where each such set contains a single vertex. Like graphs, hypergraphs can also be weighted. Let us look at some examples. In a coauthorship network, the authors are the vertices, and a set of authors constitutes a hyperedge when...

Ricci curvature of random and empirical directed hypernetworks

“…The generalizations of Forman and Ollivier Ricci curvatures introduced in Leal et al (2018) and Eidi and Jost (2020) detect complementary aspects of the connectivity patterns of directed hyperedges. The former detects the difference of the flow in the direction of the hyperedge under consideration and its size.…”

“…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).…”

Coarse Ricci curvature of hypergraphs and its generalization

“…Hence, the similar definitions of coarse Ricci curvatures for F as in Section 3 and the similar theorems as in Section 5 hold. Comparing properties of curvatures for these examples with them of other curvatures introduced by [31], [14] is an interesting problem. We leave it for a future work.…”

Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution

“…, δ → 0, where we have used the shorthand δ = d (p, q) as a shorthand to designate the distance between the two centers p and q. From here, we can build on the prior work of Forman [101], Ollivier [12][102] [103], and Eidi and Jost [104], amongst several others, in order to extend these purely geometrical definitions on Riemannian manifolds (M, g) to the more general case of arbitrary metric-measure spaces (X, d) (which include, as an important special case, directed hypergraphs). In this generalization, we replace the notion of a Riemannian metric volume element µ g by a probability measure m x , and we generalize the concept of average distance between corresponding Infinitesimal geodesic "tubes" of varying radii (1, 3 and 5, respectively) embedded in the hypergraph obtained by the evolution of the set substitution system {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}}.…”