  Irreversible opinion spreading phenomena are studied on small-world networks
generated from 2D regular lattices by means of the magnetic Eden model, a
nonequilibrium kinetic model for the growth of binary mixtures in contact with
a thermal bath. In this model, the opinion or decision of an individual is
affected by those of their acquaintances, but opinion changes (analogous to
spin flips in an Ising-like model) are not allowed. Particularly, we focus on
aspects inherent to the underlying 2D nature of the substrate, such as domain
growth and cluster size distributions. Larger shortcut fractions are observed
to favor long-range ordering connections between distant clusters across the
network, while the temperature is shown to drive the system across an
order-disorder transition, in agreement with previous investigations on related
equilibrium spin systems. Furthermore, the extrapolated phase diagram, as well
as the correlation length critical exponent, are determined by means of
standard finite-size scaling procedures.
