We study the mean traversal time $\tau$ 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 $\tau$ 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.

• Received 14 July 2004

DOI:https://doi.org/10.1103/PhysRevE.72.056119