Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Comment on "Localization Transition of Biased Random Walks on Random
  Networks"   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.

13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
2
The categories of this paper are: math
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org