  We study the Axelrod's cultural adaptation model using the concept of cluster
size entropy, $S_{c}$ that gives information on the variability of the cultural
cluster size present in the system. Using networks of different topologies,
from regular to random, we find that the critical point of the well-known
nonequilibrium monocultural-multicultural (order-disorder) transition of the
Axelrod model is unambiguously given by the maximum of the $S_{c}(q)$
distributions. The width of the cluster entropy distributions can be used to
qualitatively determine whether the transition is first- or second-order. By
scaling the cluster entropy distributions we were able to obtain a relationship
between the critical cultural trait $q_c$ and the number $F$ of cultural
features in regular networks. We also analyze the effect of the mass media
(external field) on social systems within the Axelrod model in a square
network. We find a new partially ordered phase whose largest cultural cluster
is not aligned with the external field, in contrast with a recent suggestion
that this type of phase cannot be formed in regular networks. We draw a new
$q-B$ phase diagram for the Axelrod model in regular networks.
