Discontinuous transitions have received considerable interest due to the uncovering that many phenomena such as catastrophic changes, epidemic outbreaks and synchronization present a behavior signed by abrupt (macroscopic) changes (instead of smooth ones) as a tuning parameter is changed. However, in different cases there are still scarce microscopic models reproducing such above trademarks. With these ideas in mind, we investigate the key ingredients underpinning the discontinuous transition in one of the simplest systems with up-down Z2 symmetry recently ascertained in [Phys. Rev. E 95, 042304 (2017)]. Such system, in the presence of an extra ingredient-the inertia- has its continuous transition being switched to a discontinuous one in complex networks. We scrutinize the role of three central ingredients: inertia, system degree, and the lattice topology. Our analysis has been carried out for regular lattices and random regular networks with different node degrees (interacting neighborhood) through mean-field theory (MFT) treatment and numerical simulations. Our findings reveal that not only the inertia but also the connectivity constitute essential elements for shifting the phase transition. Astoundingly, they also manifest in low-dimensional regular topologies, exposing a scaling behavior entirely different than those from the complex networks case. Therefore, our findings put on firmer bases the essential issues for the manifestation of discontinuous transitions in such relevant class of systems with Z2 symmetry.

We analyze the properties of the majority-vote (MV) model with an additional noise in which a local spin can be changed independently of its neighborhood. In the standard MV, one of the simplest nonequilibrium systems exhibiting an order-disorder phase transition, spins are aligned with their local majority with probability 1 − f , and with complementary probability f , the majority rule is not followed. In the noisy MV (NMV), a random spin flip is succeeded with probability p (with complementary 1 − p the usual MV rule is accomplished). Such extra ingredient was considered by Vieira and Crokidakis [Physica A 450, 30 (2016)] for the square lattice. Here, we generalize the NMV for arbitrary networks, including homogeneous [random regular (RR) and Erdös Renyi (ER)] and heterogeneous [Barabasi-Albert (BA)] structures, through mean-field calculations and numerical simulations. Results coming from both approaches are in excellent agreement with each other, revealing that the presence of additional noise does not affect the classification of phase transition, which remains continuous irrespective of the network degree and its distribution. The critical point and the threshold probability p t marking the disappearance of the ordered phase depend on the node distribution and increase with the connectivity k. The critical behavior, investigated numerically, exhibits a common set of critical exponents for RR and ER topologies, but different from BA and regular lattices. Finally, our results indicate that (in contrary to a previous proposition) there is no first-order transition in the NMV for large k.

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