Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks

Eleanor P Wiesler

math.DGApr 6, 2026

Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks

Authors: Eleanor P Wiesler

Organizations: The Department of Mathematics Harvard University Cambridge, Massachusetts

Abstract

This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature of Riemannian manifolds, including extensions of various theoretical bounds and theorems such as Bonnet-Myers and Levy-Gromov. Then we shift to results introduced by Lin-Lu-Yau on an extension of Ollivier-Ricci curvature on graphs, as well as the work of Jost-Liu on proving various combinatorial bounds for graph Ollivier-Ricci curvature. At the end of this thesis we present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature to various algorithms in network science and graph machine learning.

Figures & tables

Figure 1: Figure adapted from Michael Bronstein (Over-squashing, bottlenecks, and graph Ricci curvature) representing negative, zero, and positive Ricci curvature analogies between manifolds and graphs. Ricci curvature captures geodesic dispersion, which can be modeled by assessing the behavior of parallel geodesics.

Panel of Figure 2 — Figure 2: Mass transport in Monge Formulation vs. Kantorovich Relaxation

Figure 3: Neighborhood of $u$ in green, neighborhood of $v$ in blue. Each neighboor is endowed with measures $m_{u}$ and $m_{v}$ respectively for a graph $G$ .

Figure 4: Remark -1.0.32 illustration: (a) positive Ricci curvature, (b) negative Ricci curvature, (c) zero Ricci curvature.

Figure 5: Measures $m_{x}$ and $m_{y}$ function as analogues to $S_{x}$ and $S_{y}$ on a Riemannian manifold. The sign of Ollivier-Ricci curvature (ORC) depends solely on the difference between $d(x,y)$ and $W_{1}(m_{x},m_{y})$ . In particular, if if $W_{1}(m_{x},m_{y})<d(x,y)$ then ORC is positive, $W_{1}(m_{x},m_{y})>d(x,y)$ ORC is negative, and if $W_{1}(m_{x},m_{y})=d(x,y)$ ORC is zero.

Figure 6: An example type of graph diagram used for transport plan proofs by Jost–Liu in which for an edge $xy$ , vertices $x$ and $y$ also share a common neighbor $z$ to form a triangle, and where $x$ and $y$ both have non-shared neighbors. The emphasis of the triangle is used in such proofs to include positive curvature contributions of triangles.

Figure 7: Example graphs with Ollivier–Ricci curvature computed and colored on edges. Here, $\alpha=0$ refers to zero probability of a random walk being stationary at the starting vertex.

Figure 8: Tree graphs with Ollivier–Ricci curvature computed and colored on edges. Here, $\alpha=0$ refers to zero probability of a random walk being stationary at the starting vertex.

Figure 9: Example graphs with Ollivier–Ricci curvature computed and colored on edges. Here, $\alpha=0.5$ refers to $0.5$ probability of a random walk being stationary at the starting vertex.

Figure 10: Tree example graphs with Ollivier–Ricci curvature computed and colored on edges. Here, $\alpha=0.5$ refers to $0.5$ probability of a random walk being stationary at the starting vertex.

Figure 11: Directionality of edges can increase complexity with regards to mass transport or flow through a vertex, resulting in more possible transport plans and potential for mass blockage, bottlenecks, or sinks. (a) Continuous directed flow from $\hat{e}$ to $e$ , (b) Continuous directed flow from $e$ to $\hat{e}$ , (c) Restriction to only inward flow to $v_{1}$ , (d) Restriction to only outward flow from $v_{1}$ .

Figure 14: Bidirected graph, bidirected multigraph, and mixed graph. We will discuss primarily only simple directed graphs in this section, but directed networks can be of any number of these forms.

Panel of Figure 15 — Figure 15: Effective length 3 (left) through 1 (right) for directed triangles.

Panel of Figure 16 — Figure 16: Variety of effective length values for directed quadrangles.

Figure 17: Four possible variations of a directed and oriented 4-cycle, adding more possible cyclic variations compared with the undirected case.

Figure 18: Mass distribution rules must change from Jost and Liu.

Figure 19: Directed out-branching tree (above) and in-branching tree (below).

Figure 20: From Sia et al. [ SiaJayson2019OCMt ] , who use an Ollivier–Ricci curvature-based community detection algorithm to uncover community structures in an undirected drug–drug interaction dataset obtained from the DrugBank v4.1 database.

Figure 21: Figure by Emmanuel Abbe [ abbe2018community ] . Left and right graphs are identical. These two graphs have been created with the SBM model with 1000 vertices and 5 specified communities (balanced), intra-community probability of 1/50 and inter-community probability of 1/1000 for edge formation.

Figure 22: Example of a directed stochastic block model with 50 nodes, specified inter- and intra-community edge probabilities, and Ollivier–Ricci curvature plotted on edges.

w^{t}(\{u,v\})\leftarrow(1-\nu\cdot\kappa(\{u,v\}))d_{G}(u,v)

Algorithm 2 Curvature-based Clustering via Ricci Flow from [ TianYu2023CCoG ]

Figure 23: From [ topping2021understanding ] : the top of this figure demonstrates how reduction of a bottleneck may be prevented through increased volume. The paper shows that bottlenecks can be reduced on graphs by adding edges between communities of positive curvature for improved message passing in graph neural networks. Negative Ricci curvature is shown in blue and positive Ricci curvature in red.

Figure 24: Graph rewiring in the directed case must consider directionality of added edges when reducing over-smoothing and over-squashing.

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A Residual-Shell-Based Lower Bound for Ollivier-Ricci Curvature

Xiang Gu, Huichun Zhang, Jian Sun

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China · School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, Guangdong, China

Ollivier-Ricci Curvature Approximation Algorithms

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Ollivier-Ricci curvature (ORC), defined via the Wasserstein distance that captures rich geometric information, has received growing attention in both theory and applications. However, the high computational cost of Wasserstein distance evaluation has significantly limited the broader practical use of ORC. To alleviate this issue, previous work introduced a computationally efficient lower bound as a proxy for ORC based on 1-hop random walks, but this approach empirically exhibits large gaps from the exact ORC. In this paper, we establish a substantially tighter lower bound for ORC than the existing lower bound, while retaining much lower computational cost than exact ORC computation, with practical speedups of tens of times. Moreover, our bound is not restricted to 1-hop random walks, but also applies to k-hop random walks (k > 1). Experiments on several fundamental graph structures demonstrate the effectiveness of our bound in terms of both approximation accuracy and computational efficiency.

Abstract

Ollivier-Ricci Curvature

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