Potts model with invisible states on a scale-free network

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 $r$.