  We consider the general p-state Potts model on random networks with a given
degree distribution (random Bethe lattices). We find the effect of the
suppression of a first order phase transition in this model when the degree
distribution of the network is fat-tailed, that is, in more precise terms, when
the second moment of the distribution diverges. In this situation the
transition is continuous and of infinite order, and size effect is anomalously
strong. In particular, in the case of $p=1$, we arrive at the exact solution,
which coincides with the known solution of the percolation problem on these
networks.
