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

– 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 smallworld 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. Complex networks are known to underly a wide class of natural and artificial systems, ranging from cellular tissues and human populations to language structures and computer webs [1, 2, 3, 4]. Among the various mathematical models for complex networks, small-world networks [1] capture a key property of actual biological and social systems, namely, the fact that sites with distant locations in space may be separated by only a few links over the network. Small-world networks, in fact, are constructed by adding randomly distributed shortcuts between distant sites in an initially ordered lattice, such that the average path length between any two sites results in a logarithmic dependence on the network size [5,6]. The effect of the underlying geometry on the dynamics of processes occurring on small-world networks has been analyzed with particular emphasis on propagation phenomena, such as diffusion and percolation [7, 8, 9, 10, 11, 12]. It has been shown that, in most cases, any finite density of shortcuts induces the behavior expected for random networks, where diffusion is a highly efficient transport mechanism and the usual linear scaling between mean square displacement and time breaks down. On the other hand, for slightly more complicated dynamics –where diffusion is combined with certain reaction-like processes– the random-network regime occurs only above a critical (finite) shortcut density [11, 12]. In all cases, however, the relevant quantities depend monotonically on the shortcut density. This is also the case for other kinds of phenomena on small-world networks, such as for Ising dynamics [5]. In this letter, we introduce a diffusion process on a small-world network with directed shortcuts where, in contrast, non-monotonic behavior is observed as the shortcut density is varied –i.e. as the