Competition between surface relaxation and ballistic deposition models in scale-free networks

Abstract

In this paper we study the scaling behavior of the fluctuations in the steady state WS with the system size N for a surface growth process given by the competition between the surface relaxation (SRM) and the ballistic deposition (BD) models on degree uncorrelated scale-free (SF) networks, characterized by a degree distribution P(k) ∼ k−λ, where k is the degree of a node. It is known that the fluctuations of the SRM model above the critical dimension (dc = 2) scale logarithmically with N on Euclidean lattices. However, Pastore y Piontti et al. (Phys. Rev. E, 76 (2007) 046117) found that the fluctuations of the SRM model in SF networks scale logarithmically with N for λ < 3 and as a constant for λ ⩾ 3. In this letter we found that for a pure ballistic deposition model on SF networks WS scales as a power law with an exponent that depends on λ. On the other hand when both processes are in competition, we find that there is a continuous crossover between a SRM behavior and a power law behavior due to the BD model that depends on the occurrence probability of each process and the system size. Interestingly, we find that a relaxation process contaminated by any small contribution of ballistic deposition will behave, for increasing system sizes, as a pure ballistic one. Our findings could be relevant when surface relaxation mechanisms are used to synchronize processes that evolve on top of complex networks.