constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p30-c0p50 "fig:"){width="0.45\columnwidth"}   ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p50-c0p50 "fig:"){width="0.45\columnwidth"}
                                                                                                                                                                                        $p=0.8$                                                                                                                                                                                                                                                                                                                                                                              $p=1.0$
   ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p80-c0p50 "fig:"){width="0.45\columnwidth"}   ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p1p00-c0p50 "fig:"){width="0.45\columnwidth"}
  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Clearly, by setting the rewiring probability $p$ large enough, one can also recover the mean-field bifurcation diagrams as function of $J$, $K$ and $W$, with a good correspondence of the critical values of parameters.

Partial asynchronism (dilution)
-------------------------------

The dilution $d$ is the fraction of sites chosen at random that are not updated at every time step. We define the diluted 