Exact solution of mean geodesic distance for Vicsek fractals

TL;DR

This paper derives an exact formula for the mean geodesic distance in Vicsek fractals, revealing its exponential growth related to the fractal's self-similar structure and confirmed by numerical validation.

Contribution

It provides the first exact, closed-form expression for the mean geodesic distance in Vicsek fractals based on their recursive self-similar structure.

Findings

Mean geodesic distance grows exponentially with number of nodes

Exact formula derived using recurrence relations

Numerical calculations confirm the theoretical results

Abstract

The Vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important. In this paper, the exact formula for the mean geodesic distance of Vicsek fractals is found. The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractals considered. The obtained exact solution exhibits that the mean geodesic distance approximately increases as an exponential function of the number of nodes, with the exponent equal to the reciprocal of the fractal dimension. The closed-form solution is confirmed by extensive numerical calculations.