Comment on "Localization Transition of Biased Random Walks on Random Networks"

TL;DR

This paper analytically confirms the existence of a critical bias strength in biased random walks on Galton-Watson trees, clarifying the transition behavior and N dependence of the mean return time, challenging previous scaling assumptions.

Contribution

It provides an analytical calculation of the mean return time on Galton-Watson trees, confirming the transition at the critical bias and clarifying the N dependence, which contradicts earlier expectations.

Findings

Mean return time exhibits a transition at the critical bias b_c.

Analytical computation of MRT on GW trees confirms the transition.

N dependence of MRT differs from previous N scaling expectations.

Abstract

Sood and Grassberger studied in [Phys. Rev. Lett. 99, 098701 (2007)] random walks on random graphs that are biased towards a fixed target point. They put forward a critical bias strength b_c such that a random walker on an infinite graph eventually reaches the target with probability 1 when b>b_c, while a finite fraction of walks drift off to infinity for b<b_c. They rely on rigorous results obtained for biased walks on Galton-Watson (GW) trees to calculate b_c, and give arguments indicating that this result should also hold for random graphs such as Erdos-Renyi (ER) graphs and Molloy-Reed (MR) graphs. To validate their prediction, they show by numerical simulations that the mean return time (MRT) on a finite ER graph, as a function of the graph size N, exhibits a transition around the expected b_c. Here we show that the MRT on a GW tree can actually be computed analytically. This allows us (i) to show analytically that indeed the MRT displays a transition at b_c, (ii) to elucidate the N dependence of the MRT, which contradicts the \propto N scaling expected in [Phys. Rev. Lett. 99, 098701 (2007)] for b<b_c.