  Non-Markovian dynamics pervades human activity and social networks and it
induces memory effects and burstiness in a wide range of processes including
inter-event time distributions, duration of interactions in temporal networks
and human mobility. Here we propose a non-Markovian Majority-Vote model (NMMV)
that introduces non-Markovian effects in the standard (Markovian) Majority-Vote
model (SMV). The SMV model is one of the simplest two-state stochastic models
for studying opinion dynamics, and displays a continuous order-disorder phase
transition at a critical noise. In the NMMV model we assume that the
probability that an agent changes state is not only dependent on the majority
state of his neighbors but it also depends on his {\em age}, i.e. how long the
agent has been in his current state. The NMMV model has two regimes: the aging
regime implies that the probability that an agent changes state is decreasing
with his age, while in the anti-aging regime the probability that an agent
changes state is increasing with his age. Interestingly, we find that the
critical noise at which we observe the order-disorder phase transition is a
non-monotonic function of the rate $\beta$ of the aging (anti-aging) process.
In particular the critical noise in the aging regime displays a maximum as a
function of $\beta$ while in the anti-aging regime displays a minimum. This
implies that the aging/anti-aging dynamics can retard/anticipate the transition
and that there is an optimal rate $\beta$ for maximally perturbing the value of
the critical noise. The analytical results obtained in the framework of the
heterogeneous mean-field approach are validated by extensive numerical
simulations on a large variety of network topologies.
