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arXiv · 2606.27248

A Shortest-Path Anisotropy Diagnostic for Black-Hole Graph Geometries

Ariadna Uxue Palomino Ylla
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Abstract

Shortest paths in a geometric graph carry information not only about connectivity, but also about how the underlying sampled geometry is encoded by the graph. Here we study this idea for embedded spatial slices of static, spherically symmetric black hole geometries. We define a shell-based statistic, \(C_{\log}\), by measuring the cubic mean deviation of the logarithmic number of shortest paths from a reference vertex to vertices on a fixed graph-distance shell. The statistic is evaluated on graph discretizations of Schwarzschild/Flamm, Reissner--Nordstr\"om, Bardeen, and Hayward embedding geometries, together with matched-flat controls constructed from the same radial and angular samples. Within the calibrated graph-construction protocol, \(C_{\log}\) develops a reproducible radial organization: the Schwarzschild/Flamm benchmark shows a strong association with the logarithmic Kretschmann profile, and the same pattern persists across charged and regular black hole deformations over ten random seeds. The matched-flat controls do not reproduce this behavior. Tests on additional non-black-hole benchmark surfaces indicate that the response is protocol-dependent rather than universal. Thus, shortest-path multiplicity anisotropy provides a finite-shell diagnostic of curvature-organized structure in these geometric graph discretizations.

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Ariadna Uxue Palomino Ylla. 2026-06-25. A Shortest-Path Anisotropy Diagnostic for Black-Hole Graph Geometries. https://arxiv.org/abs/2606.27248

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