The precursor of the critical transitions in majority vote model with the noise feedback from the vote layer

In this paper, we investigate phase transitions in the majority-vote model coupled with noise layers of different structures. We examine the square lattice and random-regular networks, as well as their combinations, for both vote layers and noise layers. Our findings reveal the presence of independent third-order transitions in all cases and dependent third-order transitions when critical transitions occur. This suggests that dependent third-order transitions may serve as precursors to critical transitions in non-equilibrium systems. Furthermore, we observe that when the structure of vote layers is decentralized, the coupling between the vote layer and the noise layer leads to the absence of critical phenomena.