Flux fluctuations in a multi-random-walker model and surface growth dynamics

Abstract

We study the dynamics of visitation flux in a multi-random-walker model by comparison to surface growth dynamics in which one random walker drops a particle to a node at each time the walker visits the node. In each independent experiment (trial or day) for the multi-random-walker model, the number of walkers are randomly chosen from the uniform distribution $[< N_{RW} > -\triangle N_{RW}, < N_{RW} > +\triangle N_{RW} ]$. The averaged fluctuation $\bar σ ({T_{RW}})$ of the visitations over all nodes $i$ and independent experiments is shown to satisfy the power-law dependence on the walk step $T_{RW}$ as $\bar σ ({T_{RW}})\simeq {T_{RW}}^β$. Furthermore two distinct values of the exponent $β$ are found on a scale-free network, a random network and regular lattices. One is $β_i$, which is equal to the growth exponent $β$ for the surface fluctuation $W$ in one-random-walker model, and the other is $β=1$. $β_i$ is found for small $\triangle N_{RW}$ or for the system governed by the internal intrinsic dynamics. In contrast $β=1$ is found for large $\triangle N_{RW}$ or for the system governed by the external flux variations. The implications of our results to the recent studies on fluctuation dynamics of the nodes on networks are discussed.