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puted the modulus of the magnetization at the inflection point and the maximum of the magnetic susceptibility. The logarithm of these quantities as a function of the logarithm of L are presented in Figures 3 and 4, respec- tively. A linear fit of these data gives β/ν = 0 . 135(9) from the magnetization and γ/ν = 1 . 751(4) from the susceptibility which should be compared to β/ν = 0 . 125 and γ/ν = 1 . 75 obtained for a regular 2 D lattice. One can see that for this undirected random lattice we get the same universal critical behavior as the regular lat- tice model, in the same way as has been reported for the diluted spin-1 / 2 Ising model $^{32}$. 

<img src='content_image/438248.jpg'>

<img src='content_image/438251.jpg'>

## B. directed random two-dimensional lattice 

In Fig. 5 we show the dependence of the magnetization as a function of the temperature, obtained from simula- tions on directed random two-dimensional lattice, with N = 16000 sites, and two values of probability q , namely q = 0 . 1 for second-order phase transition and q = 0 . 9 for first-order phase transition, respectively. The energetic 

<img src='content_image/438249.jpg'>

Binder cumulant as a function of the reduced tempera- ture K for q = 0 . 1, and different lattice sizes, is displayed in the Figure 6, where a characteristic second-order phase transition is observed. On the other hand, in the Figure 

<img src='content_image/438250.jpg'>

7 we display the same plot as in Figure 6, but now for q = 0 . 4, where we can see that a first-order transition is present for this value of the rewiring probability. 

In Fig.8, the difference 2 / 3 − B$_{min}$ is shown as a func- tion of the parameter 1 /N for different probabilities q . 

