Spatial log-periodicity of symmetric first-passage times in finite Sierpinski graphs: geometric frequency, stability, and extrapolation

Abstract

We investigate how the mean time for a random walker to first reach a target depends on source–target separation in three finite Sierpinski graph families: planar, tetrahedral, and scale-three planar. After removing the leading growth with distance, we examine small oscillations that approximately repeat when the separation is multiplied by the geometric scale factor. These oscillations are spatial and log-periodic. Deterministic graph calculations provide symmetric first-passage means, grouped using shortest-path or Euclidean distance. In the primary shortest-path configurations, freely fitted frequencies are 0.995, 0.980, and 1.025 cycles per unit of log_lambda(r), close to the geometric prediction f = 1. Selected common-distance intervals across successive generations give relative amplitude changes of 2.7–5.2% in magnitude and phase changes below 0.027 radians. Applying parameters, including the freely estimated frequency, from one generation to larger distances in the next reduces the prediction RMSE of the normalized observable by 43.6–65.2% in the primary transfers. This improvement depends on the distance metric and interval; an earlier SG3 transfer worsens. Synthetic controls test recovery under known oscillations and expose misleading fits to nonperiodic or variable-frequency signals. The results delimit the conditions under which the log-periodic correction describes and predicts spatial first-passage means in finite graphs, without establishing a universal criterion for identifying fractality.