Biased random walk on networks

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Abstract
A method for calculating properties of biased random walks on nontrivial networks is described. It is a generalization of a method previously developed by the authors for unbiased walks. The essence of the method is the identification of certain basic types of walks. Generating functions corresponding to complicated walks are expressible in terms of the generating functions corresponding to simple walks. Moreover, we shall demonstrate that generating functions for walks on complex structures can be expressed in terms of generating functions corresponding to substructures. For example, the properties of walks on a network composed of coupled one-dimensional objects (like a resistor network) can be calculated, once the generating functions for certain walks on a one-dimensional ‘‘segment’’ have been computed. These combination rules are not as simple as the Kirchoff rules for resistor networks. Still, they are rather straightforward. The role of bias is examined. We consider the cases of a finite segment, a bent segment (in an external field), a segment with a dangling bond, and two segments in parallel. The crossover from the diffusive (weak-bias) regime to the drift (strong-bias) regime is obtained.