Traversal Times for Random Walks on Small-World Networks

TL;DR

This paper investigates how different transition rates affect the mean traversal time of random walks on small-world networks, revealing a percolation transition and developing an effective medium theory for accurate predictions.

Contribution

It introduces a model with distinct transition rates for different steps and demonstrates a percolation transition in traversal times, supported by a self-consistent effective medium theory.

Findings

Traversal time exhibits a percolation transition when f >> F.

Universal scaling behavior observed near the transition.

Effective medium theory accurately predicts traversal times across regimes.

Abstract

We study the mean traversal time for a class of random walks on Newman-Watts small-world networks, in which steps around the edge of the network occur with a transition rate F that is different from the rate f for steps across small-world connections. When f >> F, the mean time to traverse the network exhibits a transition associated with percolation of the random graph (i.e., small-world) part of the network, and a collapse of the data onto a universal curve. This transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps. We develop a simple self-consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition, as is characteristic of most effective medium theories.