  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.
