  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.
