Majority-vote model on spatially embedded networks: Crossover from mean-field to Ising universality classes

We study through Monte Carlo simulations and finite-size scaling analysis the nonequilibrium phase transitions of the majority-vote model taking place on spatially embedded networks. These structures are built from an underlying regular lattice over which directed long-range connections are randomly added according to the probability $P_{ij}\sim r^{-\alpha }$, where $r_{ij}$ is the Manhattan distance between nodes $i$ and $j$, and the exponent $\alpha$ is a controlling parameter [J. M. Kleinberg, Nature (London) 406, 845 (2000)]. Our results show that the collective behavior of this system exhibits a continuous order-disorder phase transition at a critical parameter, which is a decreasing function of the exponent $\alpha$. Precisely, considering the scaling functions and the critical exponents calculated, we conclude that the system undergoes a crossover among distinct universality classes. For $\alpha \le 3$ the critical behavior is described by mean-field exponents, while for $\alpha \ge 4$ it belongs to the Ising universality class. Finally, in the region where the crossover occurs, $3<\alpha <4$, the critical exponents are dependent on $\alpha$.

Figure 1

Phase diagram for the majority-vote model on Kleinberg's networks. The critical parameter $q_c$ decreases monotonically with the control parameter $\alpha$. The critical values remain always above the value for the regular square lattice $q_c=0.075$ [19] (dashed line).

Figure 2

Binder's cumulant calculated at the critical noise $q_c$. The bottom (red) dotted line corresponds to the mean-field value $\left(U^{*}=0.2705208\right)$ [37] and the upper (blue) dashed line to the limit for the 2D Ising universality $U^{*}=0.61069014\left(1\right)$ [22, 38]. For some data points the error bars are smaller than the symbols. A minimum of $U\left(q_c\right)$ at $\alpha =2$ is reminiscent of the optimal navigation condition with local knowledge [28]. The 2D Ising regime is recovered for $\alpha \ge 4$.

Figure 3

Logarithmic plot showing the finite-size scaling for the critical susceptibility with $\alpha =3.0$ (circles), $3.2$ (squares), $3.8$ (triangles), and $4.0$ (stars). The solid lines represent the least-squares fits to data, whose slopes correspond to the exponent $\gamma /\overline{\nu }$ (see Table 1).

Figure 4

The dependence of the critical exponents $\beta /\overline{\nu }$ and $\gamma /\overline{\nu }$ on the parameter $\alpha$ (see Table 1). Here, $\overline{\nu }=2$ for all values of $\alpha$.

Figure 5

Data collapse for the magnetization for system sizes $N=10000\text{(circles)},22500\text{(stars)},40000\text{(rectangles)},62500\text{(triangles)}$, and $90000\text{(diamonds)}$. (a) The universal curve for $\alpha =3$ is consistent with mean-field exponents: $\beta =1/2,\gamma =1,\overline{\nu }=2$. (b) For $\alpha =4$, the data collapse is obtained using Ising exponents: $\beta =1/8,\gamma =7/4,\overline{\nu }=2$.

Figure 6

Data collapse for the susceptibility at the critical point as a function of $\alpha$. The exponents used are those from our conjecture: for $\alpha \le 3,\gamma /\overline{\nu }=0.5$ (mean field), for $3<\alpha <4$ the exponents are $\alpha$-dependent (see Table 1), and for $\alpha \ge 4,\gamma /\overline{\nu }=0.875$ (2D Ising). Notice that the maximum of the fluctuations occurs at $\alpha =3$.