Discrete Curvature Models

• Discrete curvature models are rigorous, combinatorial analogues of classical curvature, assigning curvature values on graphs, meshes, and other non‐smooth structures.
• They replicate key theorems like Gauss–Bonnet and Theorema Egregium, with convergence properties linking discrete computations to smooth geometric limits.
• Applications span geometric data analysis, network science, image processing, and discrete gravity, supported by efficient computational algorithms.

Discrete curvature models provide rigorous, quantifiable analogues of the classical geometric notion of curvature in settings where the underlying space is not smooth but combinatorial or algebraic—e.g., in graphs, simplicial complexes, polygonal meshes, matrix regularizations, or discrete Markov processes. These formulations serve as a bridge between differential geometry and discrete structures, enabling the transfer of geometric theorems (such as Gauss–Bonnet, Theorema Egregium, and Chern–Gauss–Bonnet) and variational principles into combinatorial, algebraic, and computational contexts. Discrete curvature concepts now play foundational roles in diverse applications, including geometric data analysis, network science, geometry processing, image analysis, and the discretizations of gravity and general relativity.

1. Foundations and Definitions

Discrete curvature models arise by replacing the smooth manifold with a combinatorial or algebraic object and providing an assignment of curvature values at elements such as vertices, edges, faces, or cells. Key paradigms are:

• Polygonal Curvature: For a planar polygon with vertices $C_1, \ldots, C_n$, the discrete curvature at vertex $C_i$ is the turning angle:

$$\kappa^s_i = \angle(\vec{C_{i-1}C_i}, \vec{C_iC_{i+1}}) \in (-\pi, \pi)$$

The total (signed) curvature recovers $2\pi$ times the turning number.

• Polyhedral Surface Curvature: Given a polyhedral (or piecewise linear) surface, the angle defect at a vertex $v$ is

$K_v = 2\pi - \sum_{i} \alpha_i$

where $\alpha_i$ are the face angles at $v$. This is a discrete analogue of Gaussian curvature.

• Matrix Regularizations (Noncommutative Geometry): Functions on a surface $\Sigma$ are mapped to sequences of Hermitian matrices $T^{(\alpha)}(f)$ satisfying semiclassical properties, notably for products and Poisson brackets. Discrete Gaussian curvature and Euler characteristic are defined via sequences of curvature matrices and traces:

$$\chi_\alpha = \hbar_\alpha \operatorname{Tr}(\gamma K_\alpha)$$

yielding a noncommutative Gauss–Bonnet theorem (Arnlind et al., 2010).

• Piecewise Flat Manifolds (Regge Calculus): The curvature is concentrated on hinges (codimension-2 simplices or edges in 2D) and is computed as a weighted sum of the dihedral angles (angle between normals of adjoining simplices) (Conboye, 2023).
• Graph Curvature (Combinatorial and Metric Approaches):
  • Combinatorial (Angle Defect): For planar graphs or triangulations, curvature at a vertex can be the combinatorial angle defect, as above.
  • Metric Ricci Curvature: Ricci-type curvatures for graphs include:
  • Ollivier–Ricci: Based on optimal transport of local probability measures; curvature reflects contraction or expansion of mass along edges (Pouryahya et al., 2017).

  • Bakry–Émery Ricci (Emery/Bakry–Émery Curvature): Defined for each vertex via a discrete curvature–dimension inequality involving operators $\Gamma$ and $\Gamma_2$ built from the graph Laplacian (Hoffmann et al., 23 Jan 2025, Yadav et al., 26 Oct 2025):

    $$\Gamma_2(f)(v) \geq K_v \Gamma(f)(v)\quad \forall f$$

    with $$\Gamma(f,g)(v) = \frac{1}{2}[f(v)\Delta g(v) + g(v)\Delta f(v) - \Delta(fg)(v)]$$.

• Sectional Curvature on Graphs: For graphs approximating manifolds, curvature can be estimated from metric data using generalized cosine laws applied to triangles sampled in the graph (Plessis et al., 2022).
• Curvature Tensors in Discretized Gravity: Tensorial curvature in a discrete lattice can be constructed using spin connection variables and local parallel transport rules, converging to the continuum as the lattice is refined (Chamseddine et al., 2023, Chamseddine et al., 6 Sep 2024).

2. Theoretical Results and Discrete Analogues of Classical Theorems

Discrete curvature models replicate core theorems of the differential-geometric setting:

• Gauss–Bonnet Theorem: In a polyhedral or matrix-regularized setting,

$\sum_{v} K_v = 2\pi \chi(M)$

or, in the matrix case,

$$\lim_{\alpha\to\infty} \chi_\alpha = \frac{1}{2\pi} \int_\Sigma K \sqrt{g} \, du\,dv = \chi(\Sigma)$$

(Arnlind et al., 2010, Izmestiev, 13 Feb 2025).

• Chern–Gauss–Bonnet for Higher Dimensions: Discrete Lipschitz–Killing curvatures, associated to faces of codimension $2k$, sum (alternating with dimension) to the Euler characteristic in even dimensions (Izmestiev, 13 Feb 2025).
• Theorema Egregium: In the discrete context, the vertex angle deficit (intrinsic curvature) coincides with the extrinsic measure via angle jumps between face normals, mirroring the intrinsic nature of smooth Gaussian curvature.
• Variational Principles and Discrete Flows: Discrete Yamabe and Calabi flows, based on combinatorial curvatures (possibly parameterized, as $\alpha$-curvature), exhibit convergence and rigidity properties analogous to their smooth counterparts. Discrete flows are generally studied by constructing energy functionals whose gradients yield prescribed curvatures (Xu, 2018, Xu et al., 2023).
• Hierarchy and Invariants for Discrete Curves: In 2D space forms, discrete elastic and area-constrained elastic curves satisfy discrete curvature equations that generalize the smooth variational theory and admit invariance under Bäcklund transformations (Hoffmann et al., 23 Jan 2025).

3. Computational Models and Algorithms

Numerous computational approaches have been developed to calculate discrete curvature, including:

• Local Angle and Edge-based Methods: For meshes, curvature is computed via local angles, edge lengths, or via the construction of local metrics and zweibeins, sometimes solving for discrete spin connections under torsion-free conditions (Chamseddine et al., 6 Sep 2024).
• Matrix Algebra and Noncommutative Systems: In matrix regularizations, curvature matrices are assembled via commutators of embedded coordinate matrices, with explicit trace formulas bridging discrete and continuum invariants (Arnlind et al., 2010).
• Graph-based Ricci Curvature: Efficient evaluation of discrete Ricci curvature (Bakry–Émery or Ollivier types) exploits only the local connectivity structure (e.g., the 2-ball around a vertex) or the assignment of mass to neighbors and computation of the Wasserstein distance (Pouryahya et al., 2017, Hoffmann et al., 23 Jan 2025, Yadav et al., 26 Oct 2025).
• Discrete Flows and Optimization: Discrete curvature flows are typically driven by time-stepping schemes (Yamabe/Calabi flows) or variational minimization problems, frequently coupled to dynamic triangulation/surgery or re-meshing algorithms to address singularities (Xu, 2018, Glickenstein et al., 2014).
• Piecewise Flat Meshes and Hinge Angles: Curvature concentrates on hinges, with spatial averages computed via dual tessellations (e.g., Voronoi, barycentric) and averaging over intersected regions (Conboye, 2023).
• Crystalline Curvature Flows: For anisotropic mean curvature flows on a lattice, total variation energies with direction-dependent weights encode crystalline surface tension; robust evolution is achieved by iterative minimization interleaved with discrete signed-distance reinitialization (Chambolle et al., 7 Mar 2024).
• Cosine Rule-based Estimators: For graphs sampled from manifolds, metric-based estimators apply generalized cosine laws to collections of triangles to recover sectional curvature values (Plessis et al., 2022).
• Algorithmic Complexity: Some of the most recent algorithms for discrete curvature, e.g., for arbitrary triangulated surfaces, achieve efficient scaling, e.g., $\mathcal{O}(N\log N)$ for orienting meshes and solving for spin connections (Chamseddine et al., 6 Sep 2024).

4. Applications in Data Analysis, Geometry, Physics, and Learning

Discrete curvature models underpin a wide spectrum of applications:

• Geometric Machine Learning and Network Science: Curvature constitutes a core geometric invariant for graph representation learning, helping characterize bottlenecks, community structure, and diffusion properties. For example, Bakry–Émery curvature has been employed to adapt the depth of message passing in graph neural networks (GNNs) and improve robustness in the presence of complex network architectures (Yadav et al., 26 Oct 2025).
• Image Processing and Computer Vision: Discrete mean and Gaussian curvature regularizers for images (viewed as 2D surfaces) allow variational models that enhance denoising, inpainting, and edge-preserving reconstruction. Such functionals, efficiently optimized via ADMM schemes, outperform higher-order PDE-based methods in preserving geometry (Zhong et al., 2019).
• Discrete Gravity and Physics: Discrete curvature is fundamental to lattice gravity formulations (e.g., Regge calculus), enabling coordinate-free definitions of scalar curvature, curvature tensors, and the spacetime Einstein–Hilbert action for triangulated or cubulated manifolds (Izmestiev, 13 Feb 2025, Chamseddine et al., 2023, Chamseddine et al., 6 Sep 2024).
• Data Geometry and Manifold Learning: Discrete sectional curvature provides estimators for intrinsic curvature in data-derived graphs, supporting manifold learning, quantifying metric distortion, and revealing structure in high-dimensional datasets. Estimators have been validated by reconstructing, for example, the curvature of the sphere and the radius of the Earth via triangulations of sampled data (Plessis et al., 2022).
• Biological Networks: Multiple notions of discrete Ricci curvature (Ollivier, Bakry–Émery, Forman) have been applied to the analysis of transcriptional and gene interaction networks, with empirical results indicating that higher average curvature is associated with increased robustness—e.g., in cancer networks (Pouryahya et al., 2017).
• Curvature Flows and Morphogenesis: Discrete models facilitate the simulation of mean curvature and Yamabe flows on polygonal or polyhedral meshes, providing algorithms for mesh smoothing, minimal surface computation, and geometric evolution under constraints (Xu, 2018, Chambolle et al., 7 Mar 2024).

5. Theoretical and Computational Comparisons

Advances in discrete curvature have led to rich cross-model comparisons and convergence theory:

• Convergence to Smooth Geometry: Under mesh refinement and suitable approximations (e.g., in the Hausdorff metric), discrete curvature quantities converge to their smooth analogues: total mean curvature, Lipschitz–Killing curvatures, and curvature tensors (Izmestiev, 13 Feb 2025).
• Comparisons Across Models: Although definitions of discrete curvature (e.g., via transport, combinatorics, or algebraic data) differ—Ollivier–Ricci, Bakry–Émery, Forman, angle defect, commutators—there is overall agreement in numerical experiments on canonical models (graphs, surfaces, meshes). Each approach offers distinct computational and theoretical trade-offs in terms of locality, sensitivity to embedding, and metric dependence (Pouryahya et al., 2017, Plessis et al., 2022).
• Spectral and Isoperimetric Inequalities: Curvature lower bounds in discrete settings yield strong structural properties, notably Cheeger and Buser-type inequalities relating curvature, the Cheeger constant, and Laplacian spectral gap—a direct analogue of Riemannian results (Klartag et al., 2015).
• Unified Variational and Energy-based Frameworks: Discrete curvature functionals often serve as the gradient driving geometric flows (e.g., discrete Yamabe and Calabi flows), ensuring global rigidity and convergence to metrics of constant curvature, even in the presence of surgery or mesh recombination (Xu, 2018, Xu et al., 2023).

6. Recent Extensions and Future Directions

Discrete curvature models are rapidly evolving across several axes:

• Complex Cell Complexes and Higher Dimensions: Lipschitz–Killing curvature generalizations accommodate cubical complexes, high-dimensional polytopes, and networks with higher-order interactions.
• Optimal Transport and Learning: Integration of discrete Ricci curvature with information-theoretic frameworks (e.g., Information Bottleneck principles) tailors graph geometry for optimal message passing in learning systems (Fu et al., 28 Dec 2024).
• Curvature Estimators for Fractals and Irregular Data: Advanced metric-based estimators now extract geometrical invariants from fractals and nonmanifold objects, broadening applicability to non-Euclidean and data-driven contexts (Plessis et al., 2022).
• Efficient Algorithms and Local Computations: Localized definitions (e.g., Bakry–Émery curvature on the punctured 2-ball) and formulations adapted to distributed computation are being developed, enabling scalability for very large data and network instances (Yadav et al., 26 Oct 2025).
• Extensions to Non-constant Curvature and Morphogenesis: Discretizations now model non-constant Gauss curvature fields, support branching and subwrinkling phenomena, and address open problems in geometric modeling of growing or evolving surfaces (Parkinson et al., 2023).

In conclusion, discrete curvature models constitute a robust theoretical and computational framework, encapsulating core geometric structures in combinatorial, algebraic, and applied settings. By replicating foundational theorems of geometry, supporting convergence to smooth limits, and underpinning a vast spectrum of algorithmic and data-driven applications, discrete curvature is a central and unifying tool in contemporary geometric analysis and learning.