Random Walks on Stochastic Generalized Vicsek Fractal Networks: Analytic Solution and Simulations

More attention has been paid to research of various kinds of fractals due to a great number of applications in different fields over the past years. In this paper, we present stochastic generalized Vicsek fractal networks to model underlying structures on many hyperbranched polymers. Then, we consider random walks, the widely-studied dynamical behavior, on stochastic models built in order to better understand structural properties. Specifically, we analytically derive the closed-form solution to mean first-passage time for random walks, which is a quantity that allows one to better understand the underlying structure of model in question, using a more effective approach compared with the typical computational methods including spectral technique. At the same time, some other fundamental structural parameters including Kirchhoff index and fractal dimension are obtained analytically as well. The results suggest that the fractal characteristic makes the underlying structure of network more loose, and thus leads the efficiency of delivering information in a random-walk-based manner to become lower. Finally, we conduct extensive simulations to demonstrate that theoretical analysis and numerical simulations are in perfect agreement with each other.