Combinatorial Ollivier-Ricci Curvature

• Combinatorial Ollivier-Ricci Curvature is a discrete analogue of classical Ricci curvature, defined via optimal transport that measures local geometric and probabilistic attributes of graphs.
• It employs explicit combinatorial formulations and linear assignment methods to compute curvature efficiently, thereby revealing insights into clustering, spectral gaps, and rigidity in networks.
• The approach extends to hypergraphs, directed and weighted graphs, and serves as a bridge between discrete and continuous geometry with applications in network analysis and community detection.

Combinatorial Ollivier-Ricci Curvature is a discrete analogue of Ricci curvature, defined via optimal transport on metric graphs and related combinatorial structures. This curvature quantifies how “neighborhood balls” around adjacent nodes are positioned relative to the underlying graph metric, and captures fundamental geometric and probabilistic information about the structure. The curvature has immediate connections to Markov random walks, local clustering, mixing times, spectral gaps, extremal combinatorics, and rigidity phenomena in graph theory. The precise combinatorial foundations and algorithmic aspects enable the analysis of both finite graphs (including highly regular and distance-regular examples) and random or geometric networks, facilitating connections to their smooth manifold counterparts.

1. Foundational Definition and Combinatorial Formulation

Let $G = (V, E, w)$ be a (possibly weighted) undirected graph equipped with edge-weights $w_{xy} > 0$. For each node $x \in V$, associate a local probability measure $m_x$, typically the normalized probability of a 1-step random walk:

$$m_x(y) = \begin{cases} 1/\deg(x), & y \in N(x) \ 0, & \text{otherwise} \end{cases}$$

Define $d_G(x,y)$ to be the shortest-path metric with respect to weights. The 1-Wasserstein distance (“earth-mover distance”) between $m_x$ and $m_y$ is

$$W_1(m_x, m_y) = \inf_{\pi} \sum_{u \in N(x)} \sum_{v \in N(y)} d_G(u, v) \, \pi(u, v)$$

where $\pi$ ranges over couplings with specified marginals. The (edge) Ollivier-Ricci curvature is

$\kappa(x, y) = 1 - \frac{W_1(m_x, m_y)}{w_{xy}}$

for adjacent $x \sim y$. This quantity is nontrivially determined by combinatorial structure: it is large and positive in cliques and highly clustered graphs, negative in tree-like or bridge regions, and zero on lattice-like or cycle graphs (Jost et al., 2011, Hoorn et al., 2020, Hehl, 11 Jul 2024, Hickok et al., 6 Oct 2025).

On $d$-regular graphs, Hehl (Hehl, 11 Jul 2024) provides explicit combinatorial formulas for $\kappa(x, y)$ in terms of the number of common neighbors $\Delta(x, y)$, 4-cycle and 5-cycle structure, and the solution of a linear assignment problem (optimal matching of non-common neighbors):

$$\kappa(x,y) = \frac{1}{d} \left[ d+1 - \min_{\varphi} \sum_{z \in R_x} d(z,\varphi(z)) \right]$$

where $R_x$ and $R_y$ are the non-common neighbors of $x$ and $y$, and the minimization is over bijections $\varphi: R_x \to R_y$. This assignment problem typically yields integer assignment costs in $\{1,2,3\}$, reflecting the occurrence of 4-cycles and 5-cycles containing $(x,y)$. The optimal transport can thus be analyzed combinatorially in terms of such local subgraph configurations (Hehl, 11 Jul 2024).

2. Algorithmic and Computational Aspects

For $d$-regular graphs and other settings where neighbor sets are small, the minimal-cost assignment in the bipartite graph between $R_x$ and $R_y$ can be computed efficiently. Hehl (Hehl, 11 Jul 2024) shows that the assignment cost computation per edge is $O(d^3)$, using linear-assignment algorithms such as the $\epsilon$-auction or Hungarian methods. The basic computational pipeline consists of:

1. Assembling the $|R_x| \times |R_y|$ cost matrix with entries $d(z, w)$.
2. Solving the linear assignment (minimum-cost matching) problem.
3. Returning the curvature via the explicit formula.

For general graphs, the Wasserstein distance may be formulated as a min-cost flow or linear program of size $O(d_x d_y)$ for supports $N(x)$ and $N(y)$. In the regular case or when $|N(x)| = |N(y)|$, the transport plan is a perfect matching, and by the Birkhoff-von Neumann theorem, the optimum corresponds to a permutation matrix (DasGupta et al., 2022, Kelly, 2019, Kang et al., 22 May 2024). Fast approximate curvature proxies, such as the generalized Jaccard metric (gJC), have also been developed, providing a lower computational complexity alternative while retaining fidelity in many random graph regimes (Pal et al., 2017).

Quantum algorithms have been constructed that exploit block-encoding of the distance matrix and quantum singular value transformation (QSVT) techniques, yielding exponential speedup in the classical bottleneck of all-pairs shortest path computation for large point clouds or graphs in specific settings (Nghiem et al., 10 Dec 2025).

3. Structural and Extremal Graph Theory—Rigidity, Ricci-flatness, and Bonnet–Myers Sharpness

Combinatorial Ollivier-Ricci curvature enables the analysis of rigidity and extremal phenomena analogous to classic Riemannian results:

• Rigidity Theorems: On $d$-regular graphs, the only graphs with maximal Lin–Lu–Yau and Ollivier curvature $\kappa(x, y) = 1$ for all edges are the cocktail party graphs $K_{d+2}$ minus a perfect matching. These are characterized as the only regular graphs of diameter two with this extremal property (Hehl, 11 Jul 2024).
• Ricci-flatness: The characterization of Ricci-flat graphs (edges with $\kappa(x, y) = 0$) depends sensitively on local cycle structure. For girth $\geq 5$, only paths, cycles ($C_n$, $n \geq 6$), and infinite lines are possible (Bhattacharya et al., 2013). For girth $4$, the existence of a perfect matching between the non-common neighbors of any adjacent pair characterizes Ricci-flatness; examples include certain Cartesian products and hypercubes (Bhattacharya et al., 2013, Cushing et al., 2018).
• Bonnet–Myers and Lichnerowicz Sharpness: Graphs attaining equality in the combinatorial Bonnet–Myers inequality (diameter bounded by $2/\inf \kappa$) and Lichnerowicz-type spectral gap results are fully classified and include hypercubes $Q^n$, cocktail party graphs, Johnson graphs $J(2n, n)$, demi-cubes, the Gosset graph, and certain Cartesian products (Cushing et al., 2018).

A family of $4$-regular "bone idle" graphs (with girth $4$ and all edges satisfying $\kappa(x, y) = 0$) was constructed, but no finite $3$-regular bone idle graphs exist (Hehl, 11 Jul 2024).

4. Curvature, Local Clustering, and Curvature-Dimension Inequalities

Ollivier-Ricci curvature is closely related to local clustering in graphs. Heuristically, more triangles (or higher local clustering coefficient) along an edge lead to higher (more positive) curvature (Jost et al., 2011):

$$\kappa(x, y) \geq -\left(1 - \frac{2}{d}\right) + \frac{2}{d} \cdot \frac{|\text{common neighbors}|}{d-1}$$

Analytically, uniform lower bounds on Ollivier-Ricci curvature yield curvature-dimension type (CD) inequalities as in the Bakry–Émery–Ledoux framework:

$$\Gamma_2(f, f)(x) \geq \frac{1}{N}\left(\Delta f(x)\right)^2 + K \Gamma(f, f)(x)$$

with $K$ related to the minimal curvature and $N$ an effective dimension (Jost et al., 2011, Cushing et al., 2019). This establishes connections between curvature, spectral gap, concentration inequalities, isoperimetric constants, and mixing times (Paulin, 2014).

5. Convergence to Classical Ricci Curvature and Extensions

Discrete Ollivier-Ricci curvature on random geometric graphs converges, under suitable scaling, to Ricci curvature of underlying Riemannian manifolds, providing a rigorous bridge between combinatorial and differential geometry (Hoorn et al., 2020, Hickok et al., 6 Oct 2025):

$$\lim_{n \to \infty} \; \epsilon_n^{-2} \kappa_G(x, y) \to \frac{1}{2(n+2)} \operatorname{Ric}_x(v_{xy}, v_{xy})$$

where $G$ is a random geometric graph on $n$ samples, $\epsilon_n$ is the connectivity radius, and $v_{xy}$ is the unit tangent in the direction $x \to y$.

Additionally, the “scalar Ollivier-Ricci curvature” at a vertex, defined as the weighted sum (trace) of incident edge curvatures, converges to the classical scalar curvature under manifold sampling (Hickok et al., 6 Oct 2025):

$$\mathrm{SORC}(x) = \frac{1}{\deg(x)} \sum_{y \sim x} w(x, y)^2 \cdot \kappa_G(x, y)$$

with appropriate normalization and weight-choices to mimic the Riemannian trace.

6. Extension to Hypergraphs, Weighted/Directed Graphs, and Approximations

The Wasserstein transport-based definition of Ollivier-Ricci curvature generalizes to integer-valued metric spaces, hypergraphs, and weighted graphs. For hyperedges, the curvature aggregates the Wasserstein distances over all pairs within an edge or measures barycentric transport cost (Kang et al., 22 May 2024). Jost-Liu–style combinatorial lower bounds and approximate algorithms can be generalized, offering linear or nearly-linear time calculations even for large-scale complex networks. Directed and weighted variants are accommodated through tailored measures and distance structures.

Jaccard-based proxies (JC, gJC) offer scalable and analytically justifiable approximations to Ollivier-Ricci curvature, matching its asymptotic behavior in random graph models and providing fidelity for practical applications in clustering and anomaly detection (Pal et al., 2017).

7. Applications and Global Consequences

• Spectral Theory: Vertex-edge curvature lower bounds transfer directly to spectral gap bounds via discrete Lichnerowicz and Bonnet–Myers-type theorems. For example, the minimal edge curvature lower-bounds the spectral gap $\lambda_1$ of the normalized Laplacian (Cushing et al., 2018, Bauer et al., 2011).
• Geometric and Probabilistic Invariants: Curvature implies diameter constraints (finite positive curvature implies bounded diameter), mixing and contraction rates for Markov chains, and concentration inequalities for functions of random walks (Paulin, 2014).
• Community and Network Analysis: Combinatorial Ricci flow, updating edge weights according to local curvature, enables community detection and structural analysis in data graphs by revealing clusters and bottlenecks (Torbati et al., 1 Jan 2025).
• Graph Products and Extension: Ricci-flatness properties, curvature bounds, and spectral estimates are preserved (with modification) under certain graph products (Cartesian, tensor), while strong products may introduce negative curvature on diagonal edges (Cushing et al., 2019).
• Mathematical Physics and Discrete Geometry: Ollivier-Ricci curvature is used in discrete geometry, combinatorial gravity, polyhedral manifolds, and models of discrete space-time curvature (Loisel et al., 2014, Nghiem et al., 10 Dec 2025).

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These facts collectively situate combinatorial Ollivier-Ricci curvature as a deeply connected, efficiently computable, and structurally rich invariant in the study of discrete geometry, graph theory, and their applications to network science, statistical physics, and data analysis. For a complete treatment of explicit formulas and extremal families in regular and distance-regular graphs, see (Hehl, 11 Jul 2024) and (Cushing et al., 2018). For computational, algorithmic, and random geometric graph perspectives, see (DasGupta et al., 2022, Pal et al., 2017, Hickok et al., 6 Oct 2025, Hoorn et al., 2020), and for rigorous analysis in hypergraphs and weighted settings, (Kang et al., 22 May 2024).