Ollivier-Ricci Curvature on Graphs

• Ollivier-Ricci curvature is a discrete adaptation of classical Ricci curvature that measures the overlap of vertex neighborhoods using the Wasserstein distance.
• Local clustering and triangle counts in graphs directly affect curvature by decreasing transport costs, linking combinatorial structure to geometric insights.
• Curvature-dimension inequalities based on Ollivier-Ricci curvature yield improved spectral gap estimates and deeper understanding of network mixing and robustness.

Ollivier-Ricci curvature is a discrete generalization of the classical Ricci curvature, defined on metric measure spaces and adapted to graphs by Lin and Yau. In the graph-theoretical context, it quantifies how much the neighborhoods of adjacent vertices overlap, embedding geometric insight into discrete structures via optimal transport. This curvature has significant implications for understanding local and global properties of networks, ranging from spectral bounds to analytic inequalities and connections to combinatorial features such as cycles and local clustering.

1. Definition of Ollivier-Ricci Curvature on Graphs

In a simple (possibly weighted) graph $G = (V, E)$, with the natural graph distance ($d(x, y)$ being the length of the shortest path) and each edge assigned length 1 (unweighted case), a probability measure $m_x$ is attached to each vertex $x \in V$. For unweighted, undirected graphs, the canonical choice is

$m_x(y) = \begin{cases} \frac{1}{d_x} & \text{if %%%%4%%%% is a neighbor of %%%%5%%%%}, \ 0 & \text{otherwise}, \end{cases}$

where $d_x$ is the degree of $x$. The Ollivier-Ricci curvature $K(x, y)$ along an edge $(x, y)$ is then given by

$K(x, y) = 1 - W_1(m_x, m_y),$

where $W_1(m_x, m_y)$ denotes the Wasserstein-1 (or Earth Mover's) distance between $m_x$ and $m_y$. For adjacent vertices, this is simply $1 - W_1(m_x, m_y)$, as $d(x, y) = 1$. The Wasserstein-1 distance is computed using the Kantorovich duality: $$W_1(m_x, m_y) = \sup_{f \in 1\text{-Lip}} \left\{\sum_{z \in V} f(z) m_x(z) - \sum_{z \in V} f(z) m_y(z)\right\},$$ with the supremum over all 1-Lipschitz functions $f$ with respect to the graph metric.

Significant neighborhood overlap (i.e., many shared neighbors between $x$ and $y$) typically leads to small transport cost and thus large Ricci curvature, echoing the role of volume overlap in Riemannian geometry.

2. Connection to Triangles and Local Clustering

A central insight is that the amount of overlap between the neighborhoods of two adjacent vertices is determined by the number of triangles containing both. Define

$\#(x, y) = \text{number of triangles containing both %%%%18%%%% and %%%%19%%%%}$

which equals the number of shared neighbors. The local clustering coefficient at vertex $x$, $c(x)$, is a normalized count of triangles containing $x$: $$c(x) = \frac{2}{d_x(d_x-1)} \sum_{y \sim x} \#(x, y).$$ A high $c(x)$ implies many triangles and thus high average local overlap, increasing Ollivier-Ricci curvature.

The paper provides explicit lower bounds for Ricci curvature in terms of triangle counts. Notably, Theorem 3 asserts that

$$K(x, y) \geq 2 - (1 - \#(x, y)/(d_x \wedge d_y))_+ - (1 - \#(x, y)/(d_x \vee d_y))_+,$$

where $d_x \wedge d_y$ and $d_x \vee d_y$ denote the minimum and maximum degrees of $x$ and $y$, and $(\cdot)_+ = \max(\cdot, 0)$. When $\#(x, y) = 0$, this reduces to known bounds for trees.

This quantitatively links local clustering and cycles to curvature: networks with high clustering coefficient exhibit higher or less negative Ollivier-Ricci curvature, which influences dynamical and spectral properties.

3. Curvature–Dimension Inequalities on Graphs

Curvature–dimension inequalities form the backbone connecting discrete geometry to analysis. For the graph Laplacian

$A f(x) = \sum_{y \sim x} m_x(y) f(y) - f(x),$

one defines the gradient form $\Gamma$ and its iterated version $\Gamma_2$. A discrete curvature–dimension inequality has the form

$$\Gamma_2(f, f)(x) \geq \frac{1}{2}[A f(x)]^2 + K(x) \Gamma(f, f)(x),$$

with $K(x)$ playing the curvature role. When local clustering (and thereby Ollivier-Ricci curvature) is high, this inequality is tighter than previous forms that ignore triangles. For instance, assuming $K(x, y) \geq k > 0$ for all adjacent pairs yields

$$\Gamma_2(f, f)(x) \geq (A f(x))^2 + [2 + k (d_x \vee d_y) - 1] \Gamma(f, f)(x).$$

Consequences of such inequalities include improved spectral gap estimates for the Laplacian, and functional inequalities (Sobolev, logarithmic Sobolev), paralleling analogues in Riemannian geometry.

4. Interpretational Framework and Combinatorial Bridge

The Ollivier-Ricci framework makes explicit the combinatorial/geometric origin of discrete curvature:

• Local geometric features (neighborhood overlap) translate directly into transport cost for probability measures.
• The presence of triangles (cycles of length 3) provides "shortcuts" for mass transfer, lowering the Wasserstein distance, and raising the curvature.
• Clustering coefficients, being normalized triangle counts, thus provide an immediate quantitative link between local combinatorial complexity and global analytic properties.

This perspective enables discrete curvature notions to bridge the domains of combinatorial graph theory, geometric group theory, and spectral graph analysis.

5. Application to Network Structure and Dynamics

Higher Ollivier-Ricci curvature, as arising from abundant local clustering, impacts various network properties:

• Diffusion and Mixing: High curvature impedes the "spreading out" of probability mass, analogous to balls in positively curved Riemannian spaces, accelerating mixing of random walks.
• Spectral Gap and Eigenvalue Estimates: Via curvature–dimension inequalities, positive lower curvature translates to larger spectral gaps, enhancing robustness.
• Community Structure and Robustness: In real-world networks (e.g., social networks), clustering is high, indicating "curved" geometric structure; this supports tight community structure and rapid convergence in dynamical processes.

6. Summary Table of Key Quantities

Concept	Formula	Description
Vertex measure	$m_x(y) = \frac{1}{d_x}$ if $y \sim x$, $0$ otherwise	Mass on neighbors
OR curvature for $x \sim y$	$K(x, y) = 1 - W_1(m_x, m_y)$	Difference between 1 and transport cost
Triangle count	$\#(x, y)$	Number of triangles at edge $(x, y)$
Local clustering coefficient	$$c(x) = \frac{2}{d_x(d_x-1)} \sum_{y \sim x} \#(x, y)$$	Proportion of connected neighbor pairs
Curvature-dimension inequality	$$\Gamma_2(f, f)(x)\geq \frac{1}{2} [A f(x)]^2 + K(x)\Gamma(f, f)(x)$$	Discrete Bochner-Bakry-Émery inequality

These relationships together articulate how local graph structure—in particular, triangle abundance and clustering—shapes both geometric (curvature) and analytic (spectral gap, functional inequalities) properties.

7. Broader Significance and Extensions

The paper of Ollivier-Ricci curvature on graphs has catalyzed broad developments:

• It provides a quantitative method for linking local graph combinatorics (clustering, triangles) with global properties (spectral, mixing, diffusion).
• Discrete curvature offers a versatile tool for characterizing robustness and bottlenecks in networked systems beyond traditional degree-based measures.
• The principle generalizes naturally to weighted graphs, Markov processes, and extensions in higher-order structures such as hypergraphs.

The definition of Ollivier-Ricci curvature via optimal transport unifies perspectives from geometry, probability, and combinatorics, resulting in a discrete curvature theory with direct analytic, spectral, and applied implications for network science, graph theory, and related fields (Jost et al., 2011).