# Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature

arXiv: https://arxiv.org/abs/2608.22453 · HTML: https://mlsanity.com/paper/2608.22453/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 13 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Curvature Graph Neural Network](https://mlsanity.com/paper/2106.15762/) (2021)

## Abstract (arXiv)

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 (mathcal C) gives a holonomy (Hmathcal C), whose normalized logarithm (Omegamathcal C=-smathcal C-1operatornameLog(Hmathcal C)) 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 (Omegamathcal C) 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.