  We consider the Ising model with invisible states on scale-free networks. Our
goal is to investigate the interplay between the entropic and topological
influence on a phase transition. The former is manifest through the number of
invisible states $r$, while the latter is controlled by the network node-degree
distribution decay exponent $\lambda$. We show that the phase diagram, in this
case, is characterised by two marginal values $r_{c1}(\lambda)$ and
$r_{c2}(\lambda)$, which separate regions with different critical behaviours.
Below the $r_{c1}(\lambda)$ line the system undergoes only second order phase
transition; above the $r_{c2}(\lambda)$ - only a first order phase transition
occurs; and in-between the lines both of these phase transitions occur at
different temperatures. This behaviour differs from the one, observed on the
lattice, where the Ising model with invisible states is only characterised with
one marginal value $r_{c}\simeq 3.62$ separating the first and second order
regimes.
