Correlation between Averages Times of Random Walks On an Irregularly Shaped Objects and the Fractal Dimensions

Abstract

This study is  strongly motivated by fractal dimensions concept and dearth of relevant literatures aimed at correlating the average time of Random Walks on irregularly shaped planar objects and their corresponding fractal dimensions. Twenty selected countries maps from around the World were investigated to determine their random walk parameters and their fractal dimensions. The images (maps) were scanned, burned and saved as black and white Jpeg files. A program – visual basic 6.0 was developed for fractal dimensions and random parameters estimation. The program calculated: for each image number of boxes around the boundary of an image for 20x20 grids, and the distance covered by 100 random walkers on an image for the same number of grids. Thereafter, the program displayed a log-log graph of boundary count versus grids and log-log average time versus distance. The corresponding slope of line of best fit represents respectively the fractal dimension and random walks parameters. Preliminary investigation focusing grid size suggests 20x20 grids as non-compromising in terms of results reliability and computation expenses. The random parameters estimated based on 20x20 ranges from 1.976 to 2.995 while the estimated fractal dimensions range from 1.116 to1.212. The correlation of fractal dimensions and random walks parameter had regression value R² = 0.014. The estimated fractal dimension showed that maps (images) boundaries are statistically fractal. However, there is no correlation between their fractal dimensions and the corresponding random parameters.

KEY WORDS: Random walk, Shapes, Grids, Parameters, Dimensions