-Synchronization problems in complex networks are very often studied by researchers due to its many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, Scale Free networks with degree distribution P (k) ∼ k −λ , are widely used in research since they are ubiquitous in nature and other real systems. In this paper we focus on the surface relaxation growth model in Scale Free networks with 2.5 < λ < 3, and study the scaling behavior of the fluctuations, in the steady state, with the system size N . We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of N = N * that depends on λ: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above N * , the fluctuations decrease with λ, which means that the synchronization of the system improves as λ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size N and the exponent of the degree distribution.Since a great variety of systems can be represented by complex networks, over the last decades many researchers have studied both the topology and processes that evolve on top of these networks. Systems such as neural networks, the Internet and airlines networks [1][2][3] can be described by a set of nodes connected by links that represent a relationship between them, such as an electric impulse, friendship or air traffic. Many of these real networks were found to be characterized by a Scale Free (SF) topology, given by a degree distributionwhere k is the degree of the nodes and m ≤ k ≤ k max , where m and k max are the minimum and maximum degree respectively, and λ represents the broadness of the distribution. On most real systems, such as the World Wide Web or metabolic networks, it was found that 2 < λ < 3 [1, 2]. More recently, research has focused on dynamical processes taking place on the underlying network [4][5][6][7][8][9][10]. Particularly, many mathematical and numerical models have been elaborated to study the problem of synchronization [11][12][13][14][15][16], a phenomenon present in the behavior of many collective systems. In these processes the state of the system evolves to a p-1

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