Spontaneous symmetry breaking and discontinuous phase transition for spreading dynamics in multiplex networks
We propose a spreading model in multilayer networks and study the nature of nonequilibrium phase transition in the model. The model integrates the susceptible-infected-susceptible (or susceptible-infected-recovered) spreading dynamics with a biased diffusion process among different layers. A parameter α is introduced to control the bias of the diffusion process, such that each individual prefers to move to one layer with more infected (or recovered) neighbors for larger values of α. Using stochastic simulations and mean-field theory, we show that the type of phase transition from a disease-free phase to an endemic phase (EP) depends on the value of α. When α is small enough, the system undergoes a usual continuous phase transition as an effective spreading rate β increases, as in single-layer networks. Interestingly, when α exceeds a critical value the system shows either a hybrid two-step phase transition or a one-step discontinuous phase transition as β increases. The former contains a continuous transition between the disease-free phase and a low-prevalence EP, and a discontinuous transition between the low-prevalence EP and a high-prevalence EP. For the latter, only a discontinuous transition occurs from the disease-free phase directly to the high-prevalence EP. Moreover, we show that the discontinuous transition is always accompanied by a spontaneous symmetry breaking in occupation probabilities of individuals in each layer.