rule as $$\label{p}
s_{i}(t+1) = \begin{cases}
               1 & \text{with probability  $ (1-d)   \tau(1|h_i)$,}\\
               -1 & \text{with probability  $(1-d)   \left[1-\tau(1|h_i)\right]$,}\\
               s_{i}(t) & \text{otherwise, i.e., with probability $d$,}
               \end{cases}$$ so that for $d=0$ one has the standard parallel updating rule. One time step is defined when on the average every site of the lattice is updated once. For a system with $N$ sites, the smallest value of the dilution is $d=1/N$ and then $t_d=1/d$ updates are needed to complete one time step. If $d=1/2$, $t_d=2$, etc.

The mean-field equation corresponding to dilution is $$m(t+1) = (1-d)m(t) + d f(m(t));$$ where $f$ is the the map of Eq. . The mean-field phase diagram is reported in Fig. \[fig:mfpddJ\]. Notice that the border at $d=0$ corresponds to the horizontal line in Fig. \[fig:mfpdWJ\].

![\[fig:mfpddJ\] (Color online) Mean-field bifurcation diagram of the magnetization $m$ (dots in magenta, two initial conditions), and the entropy $\eta$ (continuous curve in green), as functions of the dilution probability $d$ with $J=-10$. ](m-f-dil-k20-J-10p00-W15p00){width="\columnwidth"}

The bifurcation diagrams and the entropy $\eta$ of the magnetization as functions of of the dilution $d$ are shown in Fig. \[bifd\] for different value of the long-range connection probability $p$. It is interesting to note the 