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neighbor outgoing links to a different site chosen at ran- dom. After repeating this process for every link, we are left with a network with a density q of random two- dimensional lattice directed links. Therefore, with this procedure every site will have exactly the same amount of undirected random two-dimensional lattice outgoing links and a varying (random) number of incoming links. 

Model - We consider now the two-dimensional spin-1 Blume-Capel model on these Poissonian random lattices. The Blume-Capel Model is a generalization of the stan- dard Ising model 10 and was originally proposed for spin- 1 to account for first-order phase transition in magnetic systems $^{11,12}$. The Hamiltonian can be written as 

where the first sum runs over all nearest-neighbor pairs of sites (points in the Voronoi construction) and the spin- 1 variables S$_{i}$ assume values ± 1 , 0. In eq. (1) J is the exchange coupling and Δ is the single ion anisotropy pa- rameter. The second sum is taken over the N spins on a D -dimensional lattice. The case where S = 1 has been extensively studied by several approximate techniques in two- and three-dimensions and its phase diagram is well established $^{11–17}$. The case S > 1 has also been investi- gated according to several procedures $^{18–24}$. 

Simulations - The simulations have been performed for Δ = 0, which is the simplest case and where we have some results in the literature to compare with. The lattices comprised different sizes with N ranging from N = 250 , 500 , 1000 , 2000 , 4000 , 8000, and 16000, where N is total umber of sites. For simplicity, the linear length of the system is defined here in terms of the size of a reg- ular two-dimensional lattice L = N 1 / $^{2}$. For each system size quenched averages over the connectivity disorder are approximated by averaging over R = 100 ( N = 500 to 4000), R = 50 ( N = 8000) and R = 25 ( N = 16000) independent realizations. For each simulation we have started with a uniform configuration of spins (the results are, however, independent of the initial configuration). We ran 2 . 52 × 10 6 Monte Carlo steps (MCS) per spin with 1 . 2 × 10 5 configurations discarded for thermaliza- tion using the “perfect” random-number generator $^{25}$. In both cases we have employed the heat bath algorithm and for every 12th MCS, the energy per spin, e = E/N , and magnetization per spin, m = ∑ $_{i}$S$_{i}$ /N , were measure and recorded in a time series file. From these thermody- namic quantities the behavior of the transition can be analyzed. 

From the series of the energy measurements we can compute, by re-weighting over a controllable temperature interval Δ T , the average energy, the specific heat, and the energy fourth-order cumulant which are respectively given by 

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where K = J/k$_{B}$T , with a normalized exchange J = 1, and k$_{B}$ is the Boltzmann constant. In the above equa- tions < ... > stands for thermodynamic averages and [ ... ]$_{av}$ for averages over the different realizations. Sim- ilarly, we can derive from the magnetization measure- ments the average magnetization, the susceptibility, and the magnetic cumulants, 

Further useful quantities involving both the energy and magnetization are their derivatives 

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In order to get the transition temperature, as well as to determine the order of the transition of this model, we apply the finite-size scaling (FSS) procedure. Initially, we search for the minima of the energy fourth-order cumu- lant given by Eq. (4). This quantity gives a qualitative, as well as a quantitative, description of the order of the transition $^{26}$. For instance, it is known27 that this param- eter takes a minimum value B$_{min}$ at the effective transi- tion temperature T$_{c}$ ( N ). One can also show 28 that for a second-order transition, lim$_{N}$ →∞ (2 / 3 − B$_{min}$ ) → 0, even at T$_{c}$ , while at a first-order transition the same limit mea- suring the same quantity is small and (2 / 3 − B$_{min}$ ) ̸ = 0. 

A more quantitative analysis can be carried out through the FSS of the energy fluctuation C$_{max}$ , the sus- ceptibility maxima χ$_{max}$ , and the minima of the Binder cumulants B , U$_{2}$ and U$_{4}$ . 

If the hypothesis of a first-order phase transition is correct, we should then expect, for large systems sizes, an asymptotic FSS behavior of the form $^{29,30}$, 

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