Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature
Authors: Hao Li, Yuhan Peng, Junwen Dong
Organizations: Department of Mathematics, National University of Singapore, Singapore. · School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. · Chern Institute of Mathematics, Nankai University, Tianjin, China.
We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric gi, and each directed edge carries a reversible metric-compatible transport Fij. The ordered product around an oriented triangular loop C gives a holonomy HC, whose normalized logarithm ΩC=−sC−1Log(HC) is used as a finite-loop curvature observation. Thus the construction discretizes the geometric principle that infinitesimal holonomy is controlled by curvature, rather than treating holonomy as a heuristic feature. Since ΩC lies in the gi-orthogonal Lie algebra, it is not itself a velocity of an SPD metric. We therefore introduce two aggregation mechanisms: a commutator with a symmetric response matrix, producing symmetric Ricci-type metric responses, and an incidence-aware covariant divergence of curvature-induced edge fluxes, reflecting the relation between trace and covariant divergence. The resulting responses are locally orthogonal-gauge equivariant and can drive exponential updates that preserve positive definiteness. We also give a reversible metric-compatible parametrization of edge transports, allowing orthogonal edge factors, loop scales, weights, and response matrices to be learned while respecting the graph geometry. Known-geometry calibrations on the unit sphere test the holonomy--curvature relation, curvature preservation under nontrivial local metric representations, and the empirical recovery of edge transports from local observations.
Figures & tables
Figure 1. Holonomy–curvature calibration on the unit sphere. Exact Levi–Civita edge transport gives the signed relation θC=AC for every oriented spherical face. With the controlled scale sC=AC , the scalar estimate KC=θC/AC equals the unit-sphere curvature to numerical precision. The flat control Fij=I produces zero holonomy.
Figure 2. Metric-compatible representation calibration. A nontrivial local SPD field changes the raw matrix representation of loop curvature, but the whitened curvature gi1/2ΩCFgi−1/2 agrees with the orthonormal-frame sphere curvature ΩCO to numerical precision.
Correspondences per edge M
1
4
16
64
256
Mean transport error ∥O−O⋆∥F
2.094×10−1
6.683×10−2
3.123×10−2
1.524×10−2
7.488×10−3
Mean whitened curvature error
1.125×101
3.024
1.404
6.883×10−1
3.437×10−1
Table 1. Synthetic learned-transport calibration on the fixed sphere mesh. Each entry is the mean over eight noise realizations. The curvature error is measured after whitening at the loop base point: ∥ΩCF−ΩC⋆∥F .
Figure 3. Synthetic recovery of non-oracle edge transports. Per-edge SO(2) Procrustes estimates use only noisy vector correspondences. As the number M of correspondences increases, both the learned transport error and the induced whitened holonomy-curvature error decrease. The dashed line is the flat connection control Oij=I .
Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, and we use SRC for curvature-guided edge pruning aimed at preserving manifold structure. Overall, SRC provides a transport-based foundation for scalable curvature-driven graph transformation and manifold-oriented pruning.
Kyoichi Iwasaki, Tam Le, Hideitsu Hino
The Graduate University for Advanced Studies, SOKENDAI · The Institute of Statistical Mathematics
Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances. We introduce Entropic Curvature, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics. We define a tractable Weak Entropic Curvature proxy that lower-bounds the global entropic curvature, and from it derive (i) a Poincare-type inequality controlling oversmoothing, (ii) a transport-entropy generalization bound, and (iii) an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, unifying oversmoothing and oversquashing as opposite ends of a single curvature spectrum. We translate the theory into three practical mechanisms, the E-Gate aggregator, the ENT structural encoding, and Midpoint-Completion Rewiring (MCR), and benchmark them against SDRF, FoSR, BORF, LCP, and Graph Ricci Flow on six node-classification benchmarks, and graph-classification.
Rachid Caich, Yassine Abbahaddou
LIX, École Polytechnique IP Paris Palaiseau, France
Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from −1 to 0, while the embedding remains at curvature −1. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.