# Spontaneous symmetry breaking and discontinuous phase transition for spreading dynamics in multiplex networks

## TL;DR

This paper studies how spreading dynamics in multilayer networks can exhibit different types of phase transitions, including discontinuous ones, influenced by biased diffusion and symmetry breaking.

## Contribution

It introduces a model combining spreading with biased diffusion in multilayer networks and reveals the conditions for various phase transition types, including discontinuous transitions.

## Key findings

- Discontinuous phase transitions occur when diffusion bias exceeds a critical value.
- Spontaneous symmetry breaking accompanies the discontinuous transitions.
- The nature of the phase transition depends on the bias parameter $oldsymbol{\alpha}$.

## 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 $\alpha$ 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 $\alpha$. Using stochastic simulations and mean-field theory, we show that the type of phase transition from a disease-free phase to an endemic phase depends on the value of $\alpha$. When $\alpha$ is small enough, the system undergoes a usual continuous phase transition as an effective spreading rate $\beta$ increases, as in single-layer networks. Interestingly, when $\alpha$ exceeds a critical value the system shows either a hybrid two-step phase transition or a one-step discontinuous phase transition as $\beta$ increases. The former contains a continuous transition between the disease-free phase and a low-prevalence endemic phase, and a discontinuous transition between the low-prevalence endemic phase and a high-prevalence endemic phase. For the latter, only a discontinuous transition occurs from the disease-free phase directly to the high-prevalence endemic phase. Moreover, we show that the discontinuous transition is always accompanied by a spontaneous symmetry breaking in occupation probabilities of individuals in each layer.