Estimate the shortest paths on fractal m-gons

The self-similar sets seem to be a class of fractals which is most suitable for mathematical treatment. The study of their structural properties is important. In this paper, we estimate the formula for the mean geodesic distance of self-similar set (denote fractal m-gons). The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractal considered. Out of result, obtained exact solution exhibits that the mean geodesic distance approximately increases as a exponential function of the number of nodes (small copies with the same size) with exponent equal to the reciprocal of the fractal dimension.

► We have derived analytically solution for the mean geodesic distance of fractal m-gons. ► The mean geodesic distance scales exponentially with number of nodes at infinite network size. ► Analytical technique could guide on related studies for deterministic fractals and network models. ► The mean geodesic distance grows exponentially with increasing size of the system. ► The mean geodesic distance play role, it has a profound impact on a variety of crucial fields.