Phase Transitions of Majority-Vote Model on Modular Networks

We investigate the phase transitions behavior of the majority-vote model with noise on a topology that consists of two coupled random networks. A parameter p is used to measure the degree of modularity, defined as the ratio of intermodular to intramodular connectivity. For the networks of strong modularity (small p), as the level of noise f increases, the system undergoes successively two transitions at two distinct critical noises, f_c1 and f_c2. The first transition is a discontinuous jump from a coexistence state of parallel and antiparallel order to a state that only parallel order survives, and the second one is continuous that separates the ordered state from a disordered state. As the network modularity worsens, f_c1 becomes smaller and f_c2 does not change, such that the antiparallel ordered state will vanish if p is larger than a critical value of p_c. We propose a mean-field theory to explain the simulation results.