  Different models are proposed to understand magnetic phase transitions
through the prism of competition between the energy and the entropy. One of
such models is a $q$-state Potts model with invisible states. This model
introduces $r$ invisible states such that if a spin lies in one of them, it
does not interact with the rest states. We consider such a model using the mean
field approximation on an annealed scale-free network where the probability of
a randomly chosen vertex having a degree $k$ is governed by the power-law
$P(k)\propto k^{-\lambda}$. Our results confirm that $q$, $r$ and $\lambda$
play a role of global parameters that influence the critical behaviour of the
system. Depending on their values, the phase diagram is divided into three
regions with different critical behaviours. However, the topological influence,
presented by the marginal value of $\lambda_c(q)$, has proven to be dominant
over the entropic influence, governed by the number of invisible states $r$.
