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, kappa =mu _0-mu _2-2mu _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.
Highlights
Many empirical networks incorporate higher order relations between elements and are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs
A positive lower bound for the Ricci curvature yields the Bonnet–Myers theorem, which bounds the diameter of the space in terms of such a lower Ricci bound, the Lichnerowicz theorem for the spectral gap of the Laplacian, a control on mixing properties of Brownian motion and the Levy–Gromov theorem for isoperimetric inequalities and concentration of measures
There have been some prior proposals for extensions of Ollivier–Ricci curvature in such a direction, but our approach is more general and, as we argue, more natural both in terms of its conceptual motivation and its range of applicability to empirical networks
Summary
Many empirical networks incorporate higher order relations between elements and are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. For obtaining an upper bound for the curvature of a hyperedge we need to control μ0 which corresponds to the stable mass at directed 3 cycles (triangles in the undirected graph case) and those vertices which are in the intersection of A and B.
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