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 b(c) exists such that most walks find the target within a finite time when b > b(c). For b < b(c), a finite fraction of walks drift off to infinity before hitting the target. The phase transition at b=b(c) 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 b(c) and verify it by large scale simulations.