  We study two variants of the modified Watts threshold model with a noise
(with nonconformity, in the terminology of social psychology) on a complete
graph. Within the first version, a noise is introduced via so-called
independence, whereas in the second version anticonformity plays the role of a
noise, which destroys the order. The modified Watts threshold model, studied
here, is homogeneous and posses an up-down symmetry, which makes it similar to
other binary opinion models with a single-flip dynamics, such as the
majority-vote and the q-voter models. Because within the majority-vote model
with independence only continuous phase transitions are observed, whereas
within the q-voter model with independence also discontinuous phase transitions
are possible, we ask the question about the factor, which could be responsible
for discontinuity of the order parameter. We investigate the model via the
mean-field approach, which gives the exact result in the case of a complete
graph, as well as via Monte Carlo simulations. Additionally, we provide a
heuristic reasoning, which explains observed phenomena. We show that indeed, if
the threshold r = 0.5, which corresponds to the majority-vote model, an
order-disorder transition is continuous. Moreover, results obtained for both
versions of the model (one with independence and the second one with
anticonformity) give the same results, only rescaled by the factor of 2.
However, for r > 0.5 the jump of the order parameter and the hysteresis is
observed for the model with independence, and both versions of the model give
qualitatively different results.
