  We study numerically the ordering process of two very simple dynamical models
for a two-state variable on several topologies with increasing levels of
heterogeneity in the degree distribution. We find that the zero-temperature
Glauber dynamics for the Ising model may get trapped in sets of partially
ordered metastable states even for finite system size, and this becomes more
probable as the size increases. Voter dynamics instead always converges to full
order on finite networks, even if this does not occur via coherent growth of
domains. The time needed for order to be reached diverges with the system size.
In both cases the ordering process is rather insensitive to the variation of
the degreee distribution from sharply peaked to scale-free.
