Efficiency of long-range navigation on Treelike fractals

To get a deep understanding of a diffusion process and realizing the most efficient methods for investigating a real network, has always been of great interest and utility to us. In this work, we aim to study and compare our mobility in a network using a normal random walk and a long-range navigation strategy such as Lévy Walk. We study the Global Mean First Traverse Distance (GMFTD) and the entropy rate for this long-range navigation process and later, compare with a normal walk strategy. For this study, GMFTD is utilized instead of Global Mean First Passage Time (GMFPT). The reason for such a choice is the fact that for more accurate and precise results, we need to also take into account the cost of a large step in case of a long-range navigation strategy. Next, we continue by calculating the entropy rate in both procedures and we derive the corresponding expressions of these quantities on a Treelike fractal. Eventually, we show that while GMFTD decreases in a long-range navigation system, entropy rate will increase. We perform an analytical comparison between the two cases and clearly demonstrate the superiority of a Lévy Walk strategy. Eventually, our simulation results, based on the derived expressions, give us a very reliable estimation for the optimum value of exponent parameter, α which are in strong agreement with each other for both cases of a minimum GMFTD and a maximum entropy rate.