Ising model with spins and 1 on directed and undirected Erdös–Rényi random graphs

Using Monte Carlo simulations, we study the Ising model with spin S = 1 / 2 and 1 on directed and undirected Erdös–Rényi (ER) random graphs, with z neighbors for each spin. In the case with spin S = 1 / 2 , the undirected and directed ER graphs present a spontaneous magnetization in the universality class of mean field theory, where in both directed and undirected ER graphs the model presents a spontaneous magnetization at p = z / N ( z = 2 , 3 , … , N ) , but no spontaneous magnetization at p = 1 / N which is the percolation threshold. For both directed and undirected ER graphs with spin S = 1 , we find a first-order phase transition for z = 4 and 9 neighbors.