Abstract
In this contribution, we represent hypergraphs as partially ordered sets or posets, and provide a geometric framework based on posets to compute the Forman–Ricci curvature of vertices as well as hyperedges in hypergraphs. Specifically, we first provide a canonical method to construct a two-dimensional simplicial complex associated with a hypergraph, such that the vertices of the simplicial complex represent the vertices and hyperedges of the original hypergraph. We then define the Forman–Ricci curvature of the vertices and the hyperedges as the scalar curvature of the associated vertices in the simplicial complex. Remarkably, Forman–Ricci curvature has a simple combinatorial expression and it can effectively capture the variation in symmetry or asymmetry over a hypergraph. Finally, we perform an empirical study involving computation and analysis of the Forman–Ricci curvature of hyperedges in several real-world hypergraphs. We find that Forman–Ricci curvature shows a moderate to high absolute correlation with standard hypergraph measures such as eigenvector centrality and cardinality. Our results suggest that the notion of Forman–Ricci curvature extended to hypergraphs in this work can be used to gain novel insights on the organization of higher-order interactions in real-world hypernetworks.
Highlights
In this contribution, we represent hypergraphs as partially ordered sets or posets, and provide a geometric framework based on posets to compute the Forman–Ricci curvature of vertices as well as hyperedges in hypergraphs
Following the above mentioned properties of hypergraphs and posets, we provide a canonical method to construct two-dimensional simplicial complexes associated with hypergraphs
We shall obtain the same Euler characteristic irrespective of the model of hypergraph that we choose to operate with: the poset model P, its associated complex ∆(P ), the geometric view of posets as simplicial complexes attached to each subset of cardinality k, or the more general polyhedral model that we considered in Ref. [15]
Summary
Networks provide a powerful framework for modelling interactions within systems composed of a large number of components [1,2,3]. A fundamental limitation of all network representations is that they can only be used to model pairwise interactions within a system, whereas many real-world complex systems display higher-order interactions between three or more components [4,5]. One possible framework to model such higher-order interactions within complex systems is through the use of hypergraphs [6]. A notable direction of research in network geometry involves the study of discrete Ricci curvatures for complex networks [12]. Two-dimensional simplicial complexes are endowed with intrinsic topological and geometric properties, which allow us to compute the Euler characteristic of hypergraphs. The interpretation of hypergraphs as two-dimensional simplicial complexes allows us to define the Forman–Ricci curvature for hypergraphs. We compared the Forman–Ricci curvature of hyperedges with two standard hyperedge measures, namely eigenvector centrality [17] and cardinality
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