Non-monotonic dependence on disorder in biased diffusion on small-world networks

Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche 8400 San Carlos de Bariloche, Río Negro, Argentina

Received: 8 August 2002
Accepted: 1 October 2002

Abstract

We report numerical simulations of a strongly biased diffusion process on a one-dimensional substrate with directed shortcuts between randomly chosen sites, i.e. with a small-world–like structure. We find that, unlike many other dynamical phenomena on small-world networks, this process exhibits non-monotonic dependence on the density of shortcuts. Specifically, the diffusion time over a finite length is maximal at an intermediate density. This density scales with the length in a nontrivial manner, approaching zero as the length grows. Longer diffusion times for intermediate shortcut densities can be ascribed to the formation of cyclic paths where the diffusion process becomes occasionally trapped.