Monte Carlo Renormalization Group Method to Study the First-Order Phase Transition in the Complex Ferromagnet

An improved Monte Carlo renormalization group method is established. Based on a new model, as well as the generalized Glauber dynamics, the first-order phase transition is studied in the Erdos-Renyi network, which is used to represent the complex ferromagnet. Both static and dynamic critical exponents can be evaluated simultaneously, and the static critical exponents are evaluated to be independent of the dimension of the system. Thus by using the hyperscaling relation vd* = 2-alpha, the effective dimension d* of complex networks can be found.