  We study the homogeneous symmetrical threshold model with independence
(noise) by pair approximation and Monte Carlo simulations on Watts-Strogatz
graphs. The model is a modified version of the famous Granovetter's threshold
model: with probability $p$ a voter acts independently, i.e. takes randomly one
of two states +/-; with complementary probability $1-p$, a voter takes a given
state, if sufficiently large fraction (above a given threshold r) of
individuals in its neighborhood is in this state. We show that the character of
the phase transition, induced by the noise parameter p, depends on the
threshold r, as well as graph's parameters. For r=0.5 only continuous phase
transitions are observed, whereas for r>0.5 also discontinuous phase
transitions are possible. The hysteresis increases with the average degree <k>
and the rewriting parameter beta. On the other hand, the dependence between the
width of the hysteresis and the threshold $r$ is non-monotonic. The value of r,
for which the maximum hysteresis is observed, overlaps pretty well the size of
the majority used for the descriptive norms in order to manipulate people
within social experiments. We put results obtained within this paper in a
broader picture and discuss them in the context of two other models of binary
opinions, namely the majority-vote and the q-voter model.
