Non-universal anomalous diffusion and adsorption in asymmetric random walks on hierarchical networks

We study an asymmetric random walk on a network consisting of a one-dimensional line and hierarchy of small-world links, called the Hanoi network.\footnote{"Geometry and Dynamics for Hierarchical Regular Networks," S. Boettcher, B. Goncalves, and J. Azaret, JPA 41, 335003 (2008).} Walkers are biased along the one-dimensional line, and move in the opposite direction only along the long-range links with a probability $p$. We study the mean-square displacement $\langle r^2\rangle\sim t^{\frac{2}{d_w}}$ and find that the anomalous diffusion exponent $d_w$ depends on $p$. The behavior ranges from ballistic motion ($d_w(p=0)=1$) to an adsorped state ($d_w(p_c)=\infty$). This phase transition to the adsorped state occurs at a finite $p_c<1$. We use simulations and the renormalization group to determine these properties.