On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S = 1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S =1 was seen to show a spontaneous magnetisation. In this case, a first-order phase transition for values of connectivity z = 2 and z = 7 is well defined. On these same networks the Potts model with q=3 and 8 states is now studied through Monte Carlo simulations. We also obtained for q = 3 and 8 states a first-order phase transition for values of connectivity z = 2 and z = 7 for the directed Barabási-Albert network. Theses results are different from the results obtained for the same model on two-dimensional lattices, where for q=3 the phase transition is of second order, while for q=8 the phase transition is of first-order.