written as $$\mathcal{H} (@s) = -\sum_i s'_i
        (H + J h_i  + Zh_i^2 + W h_i^3),$$ where the correspondences among coupling constants are $$\begin{split}
H &= \tilde H - \frac{1}{2} K\tilde Z,\\
J  &= K\left(\tilde J - \frac{3K-2}{6}\tilde W\right),\\
Z &= \frac{1}{2} K^2\tilde Z, \\
W &= \frac{1}{6}K^3\tilde W.\\
\end{split}$$ In the following, we shall consider pair ($J$) and four-spin ($W$) terms, *i.e.*, $H=Z=0$, in agreement with previous investigations [@bagnoli2013].

The coupling term $J$ modulates the “linear” effects of neighbors, so $J>0$ gives a conformist (ferromagnetic) behavior and $J<0$ a contrarian (antiferromagnetic) one. The term $W$ modulates the nonlinear effects of the crowd. In this way one can model the Asch effect by inserting $J<0$ (contrarian attitude) and $W>0$ (social norms).

The time evolution of the spins is given by the parallel application of the transition probabilities $\tau(s_i'|h_i)$ that gives the probability that the spin at site $i$ and time $t+1$ takes value $s_i'$ given the local field $h_i$ at time $t$, see Fig. \[plaquette\]. The local transition probability is defined by a heat bath probability $$\begin{aligned}
\label{eq:ntau}
  \tau(s'_i| h_i) &= \dfrac{1}{1+\exp(-2 s'_i (J h_i + W h_i^3))}\nonumber\\
  &=\frac{1}{2}\left[1+s'_i\tanh (J h_i + W h_i^3)\right].\end{aligned}$$

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                                                                                                                                                                                          (a)                                                                                                                                                                                                                                                                                                                                                                                (b)
   ![\[fig:tau\] (Color 