Ising model with spins S=1/2 and 1 on directed and undirected Erdos-Renyi random graphs

Using Monte Carlo simulations, we study the Ising model with spin S = 1/2 and 1 on directed and undirected Erdos-Renyi (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. (C) 2011 Elsevier B.V. All rights reserved.