Abstract
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.
Highlights
Over the past two decades, we have witnessed the power of network science on modeling dynamical processes in complex systems made of large numbers of interacting elements [1,2,3,4,5]
Using the SIS and SIR models as two paradigmatic examples of spreading dynamics, we found that the model in multilayer networks exhibits more abundant behaviors of phase transition than that in single-layer networks
For the SIS dynamics, when a biased parameter above a critical value the transition can be either continuous from the healthy phase to the low-prevalence endemic phase and be discontinuous to the highprevalence endemic phase, or be discontinuous directly from the healthy phase phase to the high-prevalence endemic phase, depending on the value of such a biased parameter and the distribution of initial positions of individuals
Summary
Over the past two decades, we have witnessed the power of network science on modeling dynamical processes in complex systems made of large numbers of interacting elements [1,2,3,4,5]. Jiang and Zhou [51] studied the effect of resource amount on epidemic control in a modified SIS model on a two-layer network, and they found that the spreading process goes through a first-order phase transition if the infection strength between layers is weak. Chen et al [53] studied the dynamics of the SIS model in socialcontact multiplex networks when the recovery of infected nodes depends on resources from healthy neighbors in the social layer. They found that as the infection rate increases the infected density varies smoothly from zero to a finite small value and suddenly jumps to a high value, where a hysteresis phenomenon was observed.
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