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

摘要

Sood and Grassberger studied in [1] 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 [2] 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.