OLLIVIER-RICCI CURVATURE FOR HYPERGRAPHS: A UNIFIED FRAMEWORK

Abstract

Bridging geometry and topology, curvature is a powerful and expressive invariant. While the utility of curvature has been theoretically and empirically confirmed in the context of manifolds and graphs, its generalization to the emerging domain of hypergraphs has remained largely unexplored. On graphs, the Ollivier-Ricci curvature measures differences between random walks via Wasserstein distances, thus grounding a geometric concept in ideas from probability theory and optimal transport. We develop ORCHID, a flexible framework generalizing Ollivier-Ricci curvature to hypergraphs, and prove that the resulting curvatures have favorable theoretical properties. Through extensive experiments on synthetic and real-world hypergraphs from different domains, we demonstrate that ORCHID curvatures are both scalable and useful to perform a variety of hypergraph tasks in practice.

1. INTRODUCTION

Hypergraphs generalize graphs by allowing any number of nodes to participate in an edge. They enable us to faithfully represent complex relations, such as co-authorship of scientific papers, multilateral interactions between chemicals, or group conversations, which cannot be adequately captured by graphs. While hypergraphs are more expressive than graphs and other relational objects like simplicial complexes, they are harder to analyze both theoretically and empirically, and many concepts that have proven useful for understanding graphs have yet to be transferred to the hypergraph setting. Curvature has established itself as a powerful characteristic of Riemannian manifolds, as it permits the description of global properties through local measurements by harmonizing ideas from geometry and topology. For graphs, graph curvature measures to what extent the neighborhood of an edge deviates from certain idealized model spaces, such as cliques, grids, or trees. It has proven helpful, for example, in assessing differences between real-world networks (Samal et al., 2018) , identifying bottlenecks in real-world networks (Gosztolai & Arnaudon, 2021) , and alleviating oversquashing in graph neural networks (Topping et al., 2022) . One prominent notion of graph curvature is Ollivier-Ricci curvature (ORC). ORC compares random walks based at specific nodes, revealing differences in the information diffusion behavior in the graph. As the sizes of edges and edge intersections can vary in hypergraphs, there are many ways to generalize ORC to hypergraphs. While some notions of hypergraph ORC have been previously studied in isolation (e.g., Asoodeh et al., 2018; Eidi & Jost, 2020; Leal et al., 2020) , a unified framework for their definition and computation is still lacking. Contributions. We introduce ORCHID, a unified framework for Ollivier-Ricci curvature on hypergraphs. ORCHID integrates and generalizes existing approaches to hypergraph ORC. Our work is the first to identify the individual building blocks shared by all notions of hypergraph ORC, and to perform a rigorous theoretical and empirical analysis of the resulting curvature formulations. We develop hypergraph ORC notions that are aligned with our geometric intuition while still efficient to compute, and we demonstrate the utility of these notions in practice through extensive experiments. Structure. After providing the necessary background on graphs and hypergraphs and recalling the definition of Ollivier-Ricci curvature for graphs in Section 2, we introduce ORCHID, our framework for hypergraph ORC, and analyze the theoretical properties of ORCHID curvatures in Section 3. We assess the empirical properties and practical utility of ORCHID curvatures through extensive experiments in Section 4, and discuss limitations and potential extensions of ORCHID as well as directions for future work in Section 5. Further materials are provided in Appendices A.1 to A.5. We assume that all our hypergraphs are multi-hypergraphs, and we drop the prefix hyper from hypergraph and hyperedge where it is clear from context.

2. PRELIMINARIES

We denote the degree of node i, i.e., the number of edges containing i, by deg(i) = |{e ∈ E | i ∈ e}|, write i ∼ j if i is adjacent to j (i.e., there exists e ∈ E such that {i, j} ⊆ e), and use N (i) (N (e)) for the neighborhood of i (e), i.e., the set of nodes adjacent to i (edges intersecting edge e). While deg(i) = | N (i)| in simple graphs and deg(i) ≥ | N (i)| in multigraphs, these relations do not generally hold for hypergraphs. Two nodes i ̸ = j are connected in H if there is a sequence of nodes i = v 1 , v 2 , . . . , v k-1 , v k = j such that v l ∼ v l+1 for all l ∈ [k]. Every such sequence is a path in H, whose length is the cardinality of the set of edges used in the adjacency relation. We refer to the length of a shortest path connecting nodes i, j as the distance between them, denoted as d(i, j). We assume that all (hyper)graphs are connected, i.e., there exists a path between all pairs of nodes. This turns H into a metric space (H, d) with diameter diam(H) := max{d(i, j) | i, j ∈ V }. (Hyper)graphs in which all nodes have the same degree k (deg(i) = k for all i ∈ V ) are called k-regular. Three properties of hypergraphs that distinguish them from graphs give rise to additional (ir)regularities. First, hyperedges can vary in cardinality, and a hypergraph in which all hyperedges have the same cardinality r (|e| = r for all e ∈ E) is called r-uniform. Second, hyperedge intersections can have cardinality greater than 1, and we call a hypergraph s-intersecting if all nonempty edge intersections have the same cardinality s (e ∩ f ̸ = ∅ ⇔ |e ∩ f | = s for all e, f ∈ E). Third, nodes can cooccur in any number of hyperedges; we call a hypergraph c-cooccurrent if each node cooccurs c times with any of its neighbors (i ∼ j ⇔ |{e ∈ E | {i, j} ⊆ e}| = c for all i, j ∈ V ). Using this terminology, simple graphs are 2-uniform, 1-intersecting, 1-cooccurrent hypergraphs. 

Given a hypergraph

H = (V, E), the unweighted clique expansion of H is G • = (V, E • ) with E • = {{i, j} | {i, G ′ = (V ′ , E ′ ) with V ′ = V ∪E and E ′ = {{i, e} | i ∈ V, e ∈ E, i ∈ e}, and we can uniquely reconstruct H from G ′ if we know which of its parts corresponds to the original node set of H. Ollivier-Ricci Curvature for Graphs Ollivier-Ricci curvature (ORC) extends the notion of Ricci curvature, defined for Riemannian manifolds, to metric spaces equipped with a probability measure or, equivalently, a random walk (Ollivier, 2007; 2009) . On graphs, which are metric spaces with the shortest-path distance d(•, •), the ORC κ of a pair of nodes {i, j} is defined as κ(i, j) := 1 -1 d(i, j) W 1 (µ i , µ j ) , and hence, κ(i, j) = 1 -W 1 (µ i , µ j ) if i ∼ j , where µ i is a probability measure associated with node i that depends measurably on i and has finite first moment, and W 1 is the Wasserstein distance of order 1, which captures the amount of



Graphs and Hypergraphs A simple graph G = (V, E) is a tuple containing n nodes (vertices) V = {v 1 , . . . , v n } and m edges E = {e 1 , . . . , e m }, with e i ∈ V 2 for all i ∈ [m]. Here, for a set S and a positive integer k ≤ |S|, S k denotes the set of all k-element subsets of S, and for x ∈ N with 0 / ∈ N, [x] = {i ∈ N | i ≤ x}. In multi-graphs, edges can occur multiple times, and hence, E = (e 1 , . . . , e m ) is an indexed family of sets, with e i ∈ V 2 for all i ∈ [m]. Generalizing simple graphs, a simple hypergraph H = (V, E) is a tuple containing n nodes V and m hyperedges E ⊆ P(V ) \ ∅, i.e., in contrast to edges, hyperedges can have any cardinality r ∈ [n]. In a multihypergraph, E = (e 1 , . . . , e m ) is an indexed family of sets, with e i ⊆ V for all i ∈ [m].

j} ⊆ e for some e ∈ E}, where two nodes are adjacent in G • if and only if they are adjacent in H. The weighted clique expansion of H is G • endowed with a weighting function w : E • → N, where w(e) = |{e ∈ E | {i, j} ⊆ e}| for each e ∈ E • , i.e., an edge {i, j} is weighted by how often i and j cooccur in edges from H. Both of these transformations are lossy, i.e., we cannot uniquely reconstruct H from G • . The unweighted star expansion of H is the bipartite graph

