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

TL;DR

This paper investigates how surface relaxation and ballistic deposition models compete on scale-free networks, revealing how their fluctuations scale with system size and how the dominant process depends on network parameters and system size.

Contribution

It introduces a detailed analysis of the scaling behavior of surface fluctuations in competing growth models on scale-free networks, highlighting the crossover from relaxation to ballistic deposition dominance.

Findings

Ballistic deposition fluctuations scale as a power law with system size, depending on network degree exponent.

A crossover from surface relaxation to ballistic deposition behavior occurs as system size increases.

Small contributions of ballistic deposition lead to ballistic-like behavior in large systems.

Abstract

In this paper we study the scaling behavior of the fluctuations in the steady state $W_S$ 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 networks (SF), characterized by a degree distribution $P(k)\sim k^{-\lambda}$, where $k$ is the degree of a node. It is known that the fluctuations of the SRM model above the critical dimension ($d_c=2$) scales logarithmically with $N$ on euclidean lattices. However, Pastore y Piontti {\it et. al.} [A. L. Pastore y Piontti {\it et. al.}, Phys. Rev. E {\bf 76}, 046117 (2007)] found that the fluctuations of the SRM model in SF networks scale logarithmically with $N$ for $\lambda <3$ and as a constant for $\lambda \geq 3$. In this letter we found that for a pure ballistic deposition model on SF networks $W_S$ scales as a power law with an exponent that depends on $\lambda$. 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.