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

• URL: http://arxiv.org/abs/2608.22453v1
• Date: Sun, 23 Aug 2026 15:03:19 GMT
• Title: Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature
• Abstract summary: We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics.<n>(mathcal C) gives a holonomy (H_mathcal C), whose normalized relation (_mathcal C=-s_mathcal C-1operatornameLog(H_mathcal C) is used as a finite-loop curvature observation.
• Score: 5.5162169444954365
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric \(g_i\), and each directed edge carries a reversible metric-compatible transport \(F_{ij}\). The ordered product around an oriented triangular loop \(\mathcal C\) gives a holonomy \(H_{\mathcal C}\), whose normalized logarithm \(Ω_{\mathcal C}=-s_{\mathcal C}^{-1}\operatorname{Log}(H_{\mathcal C})\) 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 \(Ω_{\mathcal C}\) lies in the \(g_i\)-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.

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