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

TL;DR

This study uses numerical simulations to show that biased diffusion on small-world networks has a non-monotonic relationship with shortcut density, with maximal diffusion time at intermediate densities due to cyclic trapping paths.

Contribution

It reveals a non-monotonic dependence of diffusion time on shortcut density in biased diffusion on small-world networks, highlighting the role of cyclic trapping paths.

Findings

Diffusion time peaks at intermediate shortcut densities.

Longer diffusion times scale nontrivially with network length.

Cyclic trapping paths cause delays in diffusion.

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.