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

Technical Summary: A Shortest-Path Anisotropy Diagnostic for Black-Hole Graph Geometries

Problem Statement
The representation of curvature in discrete spaces remains a foundational challenge in gravity, network science, and computational geometry. While smooth manifolds utilize tensorial objects (e.g., Riemann tensor, Kretschmann scalar) to encode curvature, graphs lack a unique continuum analogue. Existing discrete curvature notions capture disparate geometric aspects, such as transport contraction, cell incidence, or metric-ball growth. This work addresses the question of whether geometric information regarding curved spaces, specifically black-hole embeddings, is encoded in the anisotropy of shortest-path multiplicities within a graph discretization.

Methodology
The authors propose a shell-based shortest-path anisotropy statistic, denoted as $C_{\text{log}}$, applied to graph discretizations of embedded spatial slices of static, spherically symmetric black-hole geometries.

1. Graph Construction:

   • Embedded spatial slices of four black-hole families (Schwarzschild/Flamm, Reissner–Nordström, Bardeen, and Hayward) are represented as finite point clouds.
   • Points are sampled using a radial-angular prescription calibrated on the Schwarzschild/Flamm case ($N=1000$ points).
   • A geometric graph is constructed using a distance-matrix radius rule controlled by parameter $k$ (default $k=16$).
   • Matched-Flat Controls: To isolate the effect of the embedding geometry from radial sampling artifacts, a control graph is generated using identical radial and angular samples but with the embedding height profile removed ($z \to 0$).

2. The Statistic ($C_{\text{log}}$):

   • For a reference vertex $p$ and a graph-distance shell $S_{r_g}(p)$, the number of distinct shortest paths $N_{\text{geo}}(p, q)$ to each vertex $q$ on the shell is calculated.
   • A logarithmic multiplicity variable is defined: $X_{p,q} = \log N_{\text{geo}}(p, q)$.
   • The statistic is the cubic mean deviation of $X_{p,q}$ over the shell:
     $$C_{\text{log}}(p, r_g) = \left[ \frac{1}{|S_{r_g}(p)|} \sum_{q \in S_{r_g}(p)} |X_{p,q} - \bar{X}_{p, r_g}|^3 \right]^{1/3}$$
   • The analysis focuses on the radial organization of the shell-averaged $C_{\text{log}}$ and its correlation with the radial coordinate $r$ and the logarithmic Kretschmann scalar $\log K$.

Key Results
The study evaluates the diagnostic across multiple black-hole families and parameter scans (including charge $Q$ for Reissner–Nordström and regularity parameters $g$ and $\ell$ for Bardeen and Hayward).

• Robust Radial Organization: Across all tested black-hole families, $C_{\text{log}}$ exhibits a strong, stable radial organization. Specifically, there is a strong negative correlation between the radial coordinate and $C_{\text{log}}$ (approx. $-0.96$ to $-0.98$) and a strong positive correlation with the logarithmic Kretschmann profile (approx. $0.90$ to $0.94$).
• Control Validation: Matched-flat controls, which share the same sampling density but lack the curved embedding, fail to reproduce this trend (correlation with $r$ is weak and noisy, approx. $0.18 \pm 0.26$). This indicates the signal arises from the embedded geometry, not merely the radial sampling distribution.
• Parameter Stability: The signal persists across charge deformations ($Q=0$ to $0.4$) and regular black-hole deformations ($g, \ell = 0$ to $0.3$). The results remain stable over ten random seeds.
• Comparison with Baselines:
  • Forman-Ricci Curvature: Also detects a Flamm-specific radial trend but with different sign and scale.
  • Ollivier-Ricci Curvature: Fails to clearly separate the black-hole graph from the matched-flat control in this specific construction, suggesting $C_{\text{log}}$ probes a distinct geometric feature (shortest-path multiplicity anisotropy) rather than standard transport-based curvature.
• Limitations: Tests on constant-curvature surfaces (spheres, hyperbolic disks) and saddle-like surfaces (paraboloids) show that $C_{\text{log}}$ does not act as a universal pointwise curvature scalar. Its behavior is highly dependent on the graph construction protocol and the choice of control geometry.

Significance and Claims
The paper modestly claims that shortest-path multiplicity anisotropy serves as a useful computational probe for curvature-organized structure within calibrated graph discretizations of black-hole embedding geometries.

• Not a Universal Scalar: The authors explicitly state that $C_{\text{log}}$ is not a graph-independent continuum invariant nor a discrete Kretschmann scalar. Its interpretation is contingent on the specific discretization scheme and control geometry.
• Methodological Contribution: The primary contribution is the identification of a simple, shell-based graph statistic that robustly tracks the radial organization of curvature in specific black-hole embeddings, distinguishing them from flat controls where standard radial sampling might otherwise induce spurious structure.
• Scope: The results are specific to Euclidean embeddings of spatial slices. The relationship between this graph statistic and intrinsic spacetime curvature requires further analysis.

In summary, the work demonstrates that the anisotropy of shortest-path multiplicities contains geometric information about black-hole embeddings that is not captured by simple vertex degree or standard radial sampling, provided the graph construction is fixed and matched controls are employed.