  We study the dynamical and critical behavior of a model for irreversible
opinion spreading on Barab\'asi-Albert (BA) scale-free networks by performing
extensive Monte Carlo simulations. The opinion spreading within an
inhomogeneous society is investigated by means of the magnetic Eden model, a
nonequilibrium kinetic model for the growth of binary mixtures in contact with
a thermal bath. The deposition dynamics, which is studied as a function of the
degree of the occupied sites, shows evidence for the leading role played by
hubs in the growth process. Systems of finite size grow either ordered or
disordered, depending on the temperature. By means of standard finite-size
scaling procedures, the effective order-disorder phase transitions are found to
persist in the thermodynamic limit. This critical behavior, however, is absent
in related equilibrium spin systems such as the Ising model on BA scale-free
networks, which in the thermodynamic limit only displays a ferromagnetic phase.
The dependence of these results on the degree exponent is also discussed for
the case of uncorrelated scale-free networks.
