Continuous time random walks on networks.

Abstract

We study continuous time random walks on a network. We find that the steady state of the random walk on a given node is independent of the nature of the distribution of waiting times ψ(τ) and depends only on the local neighborhood of the node. The relaxation toward the steady state is, however, dependent on ψ(τ). For ψ(τ)∼[over largeτ]τ^{-1-α},0<α<1, the relaxation behavior depends on α. For α>1,ψ(τ) possesses a finite mean and the relaxation exhibits a scaling behavior independent of the specific details of ψ(τ). We find that the distribution of recurrence times for a given node possesses the same qualitative form as ψ(τ). Starting from a node i, we study the mean number of distinct nodes visited by the random walk 〈S_{i}(t)〉, and find that analogous to the relaxation behavior, its properties are dependent on the exponent α. Interestingly, 〈S_{i}(t)〉 grows algebraically with time; that is, 〈S_{i}(t)〉∼t^{α} for α<1. On the other hand, for α>1,〈S_{i}(t)〉∼Nf(t/N〈τ〉), with the scaling function f depending only on the structural details of the network of size N. In addition, for exponentially distributed waiting times, 〈S_{i}(t)〉 grows linearly with t at small times for homogeneous networks.