  The Axelrod model of cultural dissemination has been widely studied in the
field of statistical mechanics. The traditional version of this agent-based
model is to assign a cultural vector of $F$ components to each agent, where
each component can take one of $Q$ cultural trait. In this work, we introduce a
novel set of mean field master equations to describe the model for $F=2$ and
$F=3$ in complete graphs where all indirect interactions are explicitly
calculated. We find that the transition between different macroscopic states is
driven by initial conditions (set by parameter $Q$) and the size of the system
$N$, who measures the balance between linear and cubic terms in master
equations. We also find that this analytical approach fully agrees with
simulations where the system does not break up during the dynamics and a
scaling relation related to missing links reestablishes the agreement when this
happens.
