  We study an asymmetric version of the threshold model with anticonformity
under asynchronous update mode that mimics continuous time. We study this model
on a complete graph using three different approaches: mean-field approximation,
Monte Carlo simulation, and the Markov chain approach. The latter approach
yields analytical results for arbitrarily small systems, in contrast to the
mean-field approach, which is strictly correct only for an infinite system. We
show that for sufficiently large systems, all three approaches produce the same
results, as expected. We consider two cases: (1) homogeneous, in which all
agents have the same tolerance threshold, and (2) heterogeneous, in which the
thresholds are given by a beta distribution parametrized by two positive shape
parameters $\alpha$ and $\beta$. The heterogeneous case can be treated as a
generalized model that reduces to a homogeneous model in special cases. We show
that particularly interesting behaviors, including social hysteresis and
critical mass, arise only for values of $\alpha$ and $\beta$ that yield the
shape of the distribution observed in real social systems.
