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...
