  We study the dynamics of the contact-process, one of the simplest
nonequilibrium stochastic processes, taking place on a scale-free network. We
consider the network topology as annealed, i.e. all links are rewired at each
microscopic time step, so that no dynamical correlation can build up. This is a
practical implementation of the absence of correlations assumed by mean-field
approaches. We present a detailed analysis of the contact process in terms of a
Langevin equation, including explicitly the effects of stochastic fluctuations
in the number of particles in finite networks. This allows us to determine
analytically the survival time for spreading experiments and the density of
active sites in surviving runs. The fluctuations in the topological structure
induce anomalous scaling effects with respect to the system size when the
degree distribution has an "hard" upper bound. When the upper bound is soft,
the presence of outliers with huge connectivity perturbs the picture even more,
inducing an apparent shift of the critical point. In light of these findings,
recent theoretical and numerical results in the literature are critically
reviewed.
