  We study an adaptive network model driven by a nonlinear voter dynamics. Each
node in the network represents a voter and can be in one of two states that
correspond to different opinions shared by the voters. A voter disagreeing with
its neighbor's opinion may either adopt it or rewire its link to another
randomly chosen voter with any opinion. The system is studied by means of the
pair approximation in which a distinction between the average degrees of nodes
in different states is made. This approach allows us to identify two
dynamically active phases, a symmetric and an asymmetric one. The asymmetric
active phase, in contrast to the symmetric, is characterized by different
numbers of nodes in the opposite states that coexist in the network. The pair
approximation predicts the possibility of spontaneous symmetry breaking, which
leads to a continuous phase transition between the symmetric and the asymmetric
active phases. In this case, the absorbing transition occurs between the
asymmetric active and the absorbing phases after the spontaneous symmetry
breaking. Discontinuous phase transitions and hysteresis loops between both
active phases are also possible. Interestingly, the asymmetric active phase is
not displayed by the model where the rewiring occurs only to voters sharing the
same opinion, studied by other authors. Our results are backed up by Monte
Carlo simulations.
