Localization transition of biased random walks on random networks.
We study random walks on large random graphs that are biased towards a randomly chosen but fixed target node. We show that a critical bias strength bc exists such that most walks find the target within a finite time when b > bc. For b < bc, a finite fraction of walks drift off to infinity before hitting the target. The phase transition at b=bc is a critical point in the sense that quantities such as the return probability P(t) show power laws, but finite-size behavior is complex and does not obey the usual finite-size scaling ansatz. By extending rigorous results for biased walks on Galton-Watson trees, we give the exact analytical value for bc and verify it by large scale simulations.
Sood
Vishal
V
Complexity Science Group, University of Calgary, Calgary, Canada.
Grassberger
Peter
P
D003198:Computer Simulation; D008962:Models, Theoretical; D011789:Quantum Theory