Localization Transition of Biased Random Walks on Random Networks

TL;DR

This paper investigates the phase transition in biased random walks on large random networks, identifying a critical bias strength that determines whether walks typically reach a target or drift away.

Contribution

It provides an exact analytical value for the critical bias strength and characterizes the nature of the phase transition, supported by large-scale simulations.

Findings

Existence of a critical bias strength b_c for target hitting

Second order phase transition at b=b_c

Finite size effects deviate from standard scaling

Abstract

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 drifts off to infinity before hitting the target. The phase transition at b=b_c is second order, 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.