The majority-vote model with noise on Erdös-Rényi's random graphs has been studied. Monte Carlo simulations were performed to characterize the order-disorder phase transition appearing in the system. We found that the value of the critical noise parameter qc is an increasing function of the mean connectivity z of the random graph. The critical exponents β/ν, γ/ν and 1/ν were calculated for several values of z, and our analysis yielded critical exponents satisfying the hyperscaling relation with effective dimensionality equal to unity.

We investigate the majority-vote model on small-world networks by rewiring the two-dimensional square lattice. We observe that the introduction of long-range interactions does not remove the critical character of the model, that is, the system still exhibits a well-defined phase transition. However, we find that now the critical point is a monotonically increasing function of the rewiring probability. Moreover, we find that small-world effects change the class of universality of the model.

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The three-state majority-vote model with noise on Erdös–Rényi random graphs has been studied. Using Monte Carlo simulations we obtain the phase diagram, along with the critical exponents. Exact results for limiting cases are presented, and shown to be in agreement with numerical values. We find that the critical noise qc is an increasing function of the mean connectivity z of the graph. The critical exponents , and are calculated for several values of the connectivity. We also study the globally connected network, which corresponds to the mean-field limit . Our numerical results indicate that the correlation length scales with the number of nodes in the graph, which is consistent with an effective dimensionality equal to unity.

Metropolis Monte Carlo simulations of hydrogen-bonded liquids (water and methanol) were performed with the well tested effective pair potentials TIP5P and OPLS. The Coulomb contribution for the interaction potential was damped by a factor η varied from 1 to 0.49 for water and 1 to 0.15 for methanol. As a result, the networks formed by the hydrogen-bonded molecules presented interesting properties as a function of η, including small-world patterns and percolation transitions. These complex networks were analyzed by local (clustering coefficients, average degrees), semi-global (path lengths) and global (spectral densities) properties, and islands statistics. From these properties, small-world behavior was found for η in the range 0.60-0.75 for both liquids, interestingly independent of the molecular structure of the liquid. Phase transition behavior was observed for the average degrees and the clustering coefficient curves with critical values at 0.55 for water and 0.34 for methanol. Macroscopic properties such as mass density and vaporization enthalpy were also parametrically dependent on η and they presented phase transition behavior that coincides with the critical values obtained from the topological analysis. This is probably the first time that such phase transitions are observed for these quantities and shows a direct relation between macroscopic properties and topological features of hydrogen bond networks.