In 1986 Galam introduced the use of local majority rules to study voting systems as bottom-up democratic voting in hierarchical structures [1], see also the references [2–4]. Although sociophysics has been rejected by some physicists in the eighties [5], it is has become today an active and promising area of research for interdisciplinary physicists [6, 7].

In this same context, in 1982, de Oliveira [8] proposed a non-equilibrium model called Majority Vote Model (MVM). On two-dimensional lattices it shows critical phenomena with critical exponents ν, β, and γ, as for [8–10] the equilibrium Ising model [11, 12], in agree with a hypothesis of Grinstein et al. [13].

In 2012 Biswas et al. [14] proposed a kinetic model of opinion formation. This model kinetic continuous opinion dynamics (KCOD) presents mutual interactions between the individual i, j that can be both positive and negative. In this model the fraction of negative interactions is represented by a parameter p in order to characterize the different types of distributions for the mutual interactions. The results of the continuous version the KCOD model, obtained through numerical simulations indicate the existence of a universal continuous phase transition at p = p_c with exponents of mean field (νd = 2.00(1), β = 0.50(1), and γ = 1.00(1)).

Similar to this KCOD model is the one of Deffuant et al. [15], where each person i selects another person j to talk to (no lattice) and both move their opinion toward that of the other person. For the Krause-Hegselmann model [16], each person averages over the opinions of the others in the population, again no lattice. While these two models use continuous opinions, those in the Sznajd model [17] usually are Ising-like (+1 or −1) and restricted to a square lattice with four nearest neighbors. Two Sznajd people happening to agree in their opinion convince all their six neighbors of this opinion. In all these models one may start from a random distribution of opinions and then check if the opinions all converge to one consensus, two opposing camps of opinions, or a fragmentation into many opinion clusters [18–20].

Recently, Mukherjee and Chatterjee [21] studied the KCOD model on square and cubic lattices (2D and 3D). Their numerical results indicate that the same critical behavior of the KCOD model and the Ising model in the corresponding dimensions.

In this work, we studied the KCOD on two Archimedean lattices—namely, (4, 6, 12), and (4, 8²)—through extensive Monte Carlo simulations. Pictures of the (4, 6, 12) and (4, 8²) AL are shown in Figure 1. The AL are vertex transitive graphs that can be embedded in a plane such that every face is a regular polygon. Kepler showed that there are exactly 11 such graphs. The AL are labeled according to the sizes of faces incident to a given vertex. The face sizes are sorted, starting from the face for which the list is the smallest in lexicographical order. In this way, the square lattice gets the name (4, 4, 4, 4) (abbreviated to (4⁴)), honeycomb is called (6³), and Kagome is (3, 6, 3, 6). Here, we also compared our results with those of the MVM made on (3, 4, 6, 4) and (3⁴, 6) AL.

One of our goals, besides finding the critical exponents of the KCOD model on (4, 6, 12) and (4, 8²) AL, is to verify the Grinstein et al. criterion [13], for non-equilibrium stochastic spin systems with up-down symmetry on (4, 6, 12) and (4, 8²) AL belong to the same universality class as the Ising model on (4⁴) as suggested by Grinstein et al. [13]. Here, we also compared our results with those of the MVM on (4, 6, 12) and (4, 8²) [22].

The KCOD model [14] is defined by a set of individuals with continuous opinion variables o_i(t), where the opinion of a person i at time t, takes the values in the interval [−1, +1], is situated on every node of the (4, 6, 12) and (4, 8²) AL with N = 12L² sites for (4, 6, 12) and N = 4L² sites for (4, 8²). In a population of N individuals, opinions change out of pair-wise interactions via mutual influences/couplings μ_ij as:

The pairs i, j are unrestricted, meaning the original model is defined on a fully-connected graph, giving a mean-field-like limit (infinite range interactions). The pairwise interaction implies no sum over the index j. with real random couplings μ_ij. Agent i updates his/her opinion via Equation (1) by interacting with agent j and is influenced by the coupling μ_ij. The opinions are limited to −1 ≤ o_i(t) ≤ 1. If the opinion value of an individual become higher (lower) than +1 (−1), then it is set equal to +1 (−1). This bound, along with Equation (1), defines the dynamics of the model. Here, μ_ij is a continuous random variable defined in the range [−1, +1], i.e., it takes a random real value in the range [−1, 0] or ([0, 1]) with probability p or (1−p). In other words, the disorder parameter p denotes the fraction of negative pairwise interactions. The average opinion O=|∑ioi|/N measures the ordering in the system. As a function of the fraction p of negative interactions a symmetry breaking transition occurs between an ordered and a disordered phase: below a certain value p_c of the parameter p, the system is ordered (giving a non zero value of the opinion O, defined in the following), while it is disordered above p_c (O = 0).

The critical properties of model we are interested in are the order parameter O, the order parameter fluctuations (susceptibility) OF and the reduced fourth-order cumulant of the O, namely here by O₄, defined as

where 〈⋯ 〉 stands for time averages, computed at the steady states. N_run independent simulations are averaged over.

The quantities O, OF, and O₄ depend on the disorder parameter p and obey

(finite-size scaling) with β, γ, and ν as the usual critical exponents, f_o(x), f_of(x), f_o4(x) as the finite-size scaling functions and

as the scaling variable. Thus, the size dependence of O and OF gives us the exponents β/ν and γ/ν, respectively. The maximum of susceptibility also scales as L^γ/ν. Moreover, the value of p^* for which the susceptibility has a maximum scales with L as

Therefore, Equations (4c) and (5) give the exponent 1/ν. The effective dimensionality, D_eff, is given by the hyperscaling hypothesis

Monte Carlo simulations were made on (4, 6, 12) and (4, 8²) AL with various systems of size N = 768, 3,072, 12,288, 49,152, and 196,608 for (4, 6, 12) and N = 256, 1,024, 4,096, 16,384, 65,536, and for (4, 8²) AL. We use 2 × 10⁵ Monte Carlo steps (MCS) to make the system reach the steady state, and then the time averages are estimated over the next 3 × 10⁵ MCS. One MCS is accomplished after N attempts to update the opinions of agents i and j, considering the evolution Equations (1) and (2). The results are averaged over N_run (500 ≤ N_run ≤ 2, 000) independent simulation runs for each lattice and for given set of parameters (p, N).

Figure 2 displays the dependence of the opinion O, OF, and O₄ on the disorder parameter p, obtained from simulations on (4, 6, 12) and (4, 8²) AL with N ranging from N = 256 to 196,608 sites. The shape of O(p), OF, and O₄ curves, for a given value of L, indicate the occurrence of a second-order phase transition in the system. The phase transition occurs at the value of the critical disorder parameter p_c. Such critical disorder parameter p_c is estimated as the point where the curves of the Binder cumulant O₄ for different system sizes N intercept each other [23]. The corresponding value of O₄ is represented by O4*. Then, we obtain p_c = 0.086(3) and O4*=0.59(4); p_c = 0.109(3) and O4*=0.606(5) for (4, 6, 12), and (4, 8²) AL, respectively.

The results obtained from Figures 3–6 and used in Figure 7 are summarized in Table 2.

The excellent curve collapses Figure 7 for distinct system sizes corroborates our estimated values for p_c and exponent ratios β/ν, γ/ν and 1/ν.

The resulting critical exponents and disorder parameters are collected in Table 2. One can also see that the exponent ratios β/ν, γ/ν, 1/ν are similar to MVM, Table 1, and are different from the Ising model contrary to the Grinstein's hypothesis [13]. They are different from β/ν = 0.125 and γ/ν = 1.75 obtained for a d = 2 lattices, but obey hyperscaling relation (within the error bars). Equation (6) yields effective dimensionality of systems D_eff = 1.89(6) for (4, 6, 12) and D_eff = 1.76(7) for (4, 8²). The effective dimensionalities of KCOD on our two AL are close to those for MVM on (4, 6, 12) AL (D_eff = 1.78(7)) and on (4, 8²) AL (D_eff = 1.83(6)). The results of simulations are collected in Table 2.

We studied a non-equilibrium KCOD model through extensive Monte Carlo simulations on (4, 6, 12) and (4, 8²) AL. On these lattices, the KCOD model shows a second-order phase transition. Our Monte Carlo simulations suggest that the effective dimensionality D_eff is close to two, i.e., that hyperscaling relation 2β/ν + γ/ν ≈ 2 may be valid.

Finally, we remark that the critical exponents γ/ν, β/ν, and 1/ν for KCOD on (4, 6, 12) and (4, 8²) AL are similar to the MVM model on (4, 6, 12) and (4, 8²) AL [22], see Tables 1, 2. Therefore, this model not belong to the Ising universality class [24] and the hypothesis of Grinstein has been disproved.

The author confirms being the sole contributor of this work and approved it for publication.