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

Graph-based representations of curved spaces offer a way to probe how geometric information is encoded in shortest-path structure.

In this work, we introduce a shell-based shortest-path anisotropy statistic for graph discretizations of embedded spatial slices of black-hole geometries.

The statistic is based on the cubic mean deviation of the logarithm of the number of shortest paths from a reference vertex to vertices on a graph-distance shell.

We test the diagnostic on graph discretizations of static, spherically symmetric black-hole embedding geometries, including Schwarzschild/Flamm, Reissner--Nordström, Bardeen, and Hayward backgrounds.

Across these black-hole families, the statistic exhibits a stable radial organization that is strongly associated with the logarithmic Kretschmann profile, while matched-flat controls do not reproduce the same trend.

For Reissner-Nordström geometries with $M=1/2$ and charges $Q=0,\ldots,0.4$, the seed-averaged radial and curvature-profile correlations remain stable over ten random seeds.

Similar robustness is found for Bardeen and Hayward parameter scans.

Additional tests on non-black-hole benchmark surfaces indicate that the statistic is not a universal pointwise curvature scalar; rather, it is a curvature-sensitive graph diagnostic whose interpretation depends on the graph construction and control geometry.

These results suggest that shortest-path multiplicity anisotropy can provide a useful probe of curvature-organized structure in graph discretizations of black-hole embedding geometries.