  We show that there are two classes of finite size effects for dynamic models
taking place on a scale-free topology. Some models in finite networks show a
behavior that depends only on the system size N. Others present an additional
distinct dependence on the upper cutoff k_c of the degree distribution. Since
the infinite network limit can be obtained by allowing k_c to diverge with the
system size in an arbitrary way, this result implies that there are different
routes to the thermodynamic limit in scale-free networks. The contact process
(in its mean-field version) belongs to this second class and thus our results
clarify the recent discrepancy between theory and simulations with different
scaling of k_c reported in the literature.
