1. Introduction
In the last four decades, there has been a great increase in the study of the general behavior of people in a community. From a theoretical point of view, this rather complex dynamical system must, certainly, take into account that people can not only influence their neighbors but be influenced by them as well. According to this scenario, some early models have been introduced by Stauffer [
1] and Galam [
2,
3] using basically local majority rule arguments, which are also now called dynamics of opinions. Other models, just to cite some examples, are the Ising model (IM) [
4,
5], the majority-vote model (MVM) [
6], and the Biswas, Chatterjee, and Sen (BChS) [
7,
8] model. Regarding the Ising model, it is worth mentioning that, despite being originally proposed to treat magnetic systems, almost nine decades before the new ideas by Stauffer and Galam, the IM is so versatile that it can also be employed in the study of other interdisciplinary fields as well (see, for instance, Refs. [
9,
10,
11,
12]).
In general, the dynamical properties of the models mentioned above are better studied by considering an underlined regular lattice or network, in which every site of the lattice or node of the network are occupied by an individual (sometimes also called an agent). In this way, within each lattice or network, one has the proper type of mutual interactions that are defined by the corresponding model under study, e.g., Ising model, MVM, BChS model, and others.
In the particular case of the BChS model, which was introduced in 2012 and is the subject of the present study, the pair interactions are allowed to be either positive or negative. The interaction signs are modeled by a single noise parameter
q, that represents the fraction of negative interactions. In addition, the opinion variables can be made either continuous or discrete. This non-equilibrium system has been studied from Monte Carlo simulations on different regular lattices as well as different networks. Regardless of being continuous or discrete, the model presents a second-order phase transition at
, with the noise parameter
q playing the role of temperature in ordinary magnetic phase transitions. For
, the system has a non-zero order parameter, while for
, the order parameter is zero. However, contrary to what happens in ordinary magnetic transitions, the exponents and the universality class of the model depend on the particular chosen lattice or network. The reader is directed to Ref. [
13] for a detailed relation of the universality class of the BChS model on different lattices and networks. Ref. [
14] also presents a recent review of the progress already made in the study of the BChS model.
A rather more realistic situation has been proposed by Sorin Solomon [
15,
16], by considering two different lattices, where one lattice reflects one kind of environment (for instance, home) and the other lattice a different environment (for instance, workplace). In general, one labels the sites in the workplace lattice
i and a random permutation
of the order already established in this lattice will provide the sites in the home lattice. In such real situations, the neighbors at home differ from the neighbors in the workplace (except, of course, when everybody works at home). Such constructions are called Solomon networks (SNs) [
9,
17,
18], reflecting the fact that each individual is equally shared by two lattices, just as in the King Solomon biblical story. As a result, the net interaction of the relevant variables defined at site
i is a sum of the corresponding interactions of site
i with its neighboring sites on the workplace lattice, plus the interactions with the neighbor sites of
on the home lattice. Some correlation between home and workplace lattices can be introduced into the model by choosing
not completely at random. For example, people selecting a workplace closer to their homes. However, generally only random permutations have been considered in
in order not to proliferate additional theoretical parameters. The increase in the connectivity of each site
i makes the SNs close to small-world networks [
19,
20].
The majority of the studies involving SNs have employed two linear chains and two square lattices, implying one-dimensional () and two-dimensional () topologies, respectively. However, three-dimensional () lattices can be important as well, and not only from a theoretical point of view. For instance, one could make the correspondence of environments in the country as , in towns as , and in big cities, due to their building verticalization in some areas, as . This certainly implies that people living in such different environments present, as is indeed the case, different behaviors regarding the way they live and work.
The BChS model has already been studied in different lattices and networks [
13] and also in
and
SNs [
21]. Concerning the SNs, the critical exponents of the second-order phase transition indicate that the universality class of the model is dependent on the dimensionality of the lattices, as is expected from general renormalization group arguments. It should be stressed here that from the results one has from the literature it is not possible to definitely assert that a particular model on a particular network will undergo a first- or second-order transition, or even any transition at all. It is thus worthwhile treating the BChS model in
SNs, by defining two three-dimensional lattices, namely, simple cubic ones for simplicity. It will be seen that the model does indeed belong to a different universality class, in agreement with renormalization ideas. Moreover, the hyperscaling relation is satisfied for dimensions smaller than three, meaning that the upper critical dimensionality is certainly greater than or equal to four [
22].
Thus, in the present work, the BChS dynamical system on SNs has been studied through Monte Carlo simulations, allied with finite-size scaling techniques, considering three-dimensional simple cubic lattices. The plan of the paper is the following. In the next section, the BChS model in discrete opinion dynamics is defined. In this same section, the corresponding Monte Carlo simulation details and the respective
thermodynamic quantities used to obtain the critical behavior are presented. The results are discussed in
Section 3 and some concluding remarks are summarized in the last section.
3. Results and Discussion
The general behavior of the relevant quantities for this model, namely,
O,
, and
, given by Equations (
6)–(8), are, respectively, displayed in the top panels, (a), (b), and (c), of
Figure 1. The different lattice sizes used in the present work are listed in the legend. In these panels, only the lines of the data are shown for a better visualization of the behavior one has as a function of the noise parameter
q. The details of extracting the critical properties of the model from these data are discussed below.
The fourth-order Binder cumulant of the order parameter
, as a function of the disorder parameter
q close to the phase transition, is depicted in
Figure 2 for several SNs having different numbers of nodes. One can clearly see from this figure that the system undergoes a second-order phase transition, since the cumulants tend to cross at the same value, corresponding to the critical disorder parameter
[
26]. In the axes scales used in
Figure 2, one can make a rough estimate of the critical noise and the universal value of the Binder cumulant,
and
, respectively. In the above data, the computed errors are statistical ones and for questions of clarity are not shown in
Figure 2.
However, looking closely at the interception region of the cumulants in
Figure 2, it is noticed that the crossings do not occur at the same place due to still finite-size effects. This region has been magnified in the inset of
Figure 3. Considering as a reference lattice
, given by the thicker line in the inset, one can see that due to finite-size effects the value of the noise parameter at the crossings increases as the lattice size
L systematically increases. Using these crossings as estimates of
, and plotting them as a function of
, one obtains the data shown in the main graph of
Figure 3. The errors on
have been, in this case, estimated by using the crossings of
with
, where
and
are the respective errors on
and
obtained via the jackknife procedure. It is not surprising that the data are not all aligned, because quite small lattices have been considered. However, these data look similar to previous ones obtained for the Ising and Heisenberg models [
28,
29,
30,
31]. Using Equation (
13) for the larger lattices
, when one expects that the large lattice scaling regime has been reached, one obtains
and
in the thermodynamic limit. These are more precise values for the transition disorder parameter and for the cumulant of the model than the rough estimate obtained from just inspecting the crossings in
Figure 2. Larger values of
could also be considered, but the data turn out to be more disperse, making it more difficult to obtain a reasonable fit. The critical noise for the present
SNs is listed in
Table 1 together with the results for
and
networks for comparison.
Having in hand a good estimate of the critical noise probability, Equation (
9) can be used to evaluate the critical exponent ratio
by computing the order parameter at
for different lattice sizes.
Figure 4 depicts the ln–ln plot of the average opinion at the critical disorder,
, as a function of the system sizes
L considered herein. In this case, it is easy to see that, from Equation (
9), the magnitude of the slope of the linear fit gives the critical exponent ratio
. The corresponding result
is also reproduced in
Table 1 and is clearly different from the values obtained for the same model on SNs in one and two dimensions.
The very same procedure can be easily extended to the order parameter fluctuation using Equation (10).
Figure 5 displays the ln–ln plot of the order parameter fluctuation (susceptibility) at the critical disorder value,
, at the estimated
, as a function of the lattice size
L. From Equation (10) one obtains the ratio
as the slope of the linear fit to the data. The fit provides
and is listed in
Table 1 for a comparison to the values for other dimensions.
It is worthwhile to note that Equation (10) can still be used to obtain the value of the ratio
by considering the maximum value of the order parameter fluctuation,
that occurs at
. It turns out the
is close, but not the same, as
; see
Figure 1b that shows the behavior of
close to the transition region.
Figure 5 also displays the ln–ln plot of
as a function of the lattice size
L. In this case, one obtains
. The error in this exponent ratio is larger, but still comparable to that obtained from the estimate at
. Moreover, both exponent ratios are different from the previous estimates on
and
networks.
As mentioned in the last paragraph, the value of
, where the fluctuation in the order parameter displays a maximum, is different from
. For the present model,
approaches the infinite limit
from below, as is apparent in
Figure 1b. In fact, this noise probability is generally interpreted as a new
from which the transition point
is obtained by extrapolating Equation (12) to the thermodynamic limit. Of course, in this case, one needs to know the correlation length exponent
. However, if one adopts here a different path instead, and uses
for all values of
L in Equation (12), with the already estimated
, one is able to compute the exponent
as the magnitude of the slope of the linear fit to the data. Such a procedure is shown in
Figure 6, that displays the ln–ln plot of
as a function of
L. The respective correlation length critical exponent is given by
and also displayed in
Table 1.
Finally, in
Figure 1, the data collapse of the rescaled order parameter
is shown in panel (d), the rescaled fluctuation of the order parameter
in panel (e), and the rescaled reduced Binder cumulant
in panel (f), all as a function of the rescaled probability displacement
. As in the top panels, just the lines of the actual Monte Carlo data have been used for a better evaluation of the collapse and the accuracy of the critical quantities. The corresponding lattice sizes
L are listed in the legend of (d), where only larger sizes
have been considered in the data collapse. It can be seen that the data collapse presents an excellent agreement with scaling relations given in Equations (
9), (10) and (12), mainly for
and even close to
, which indicates that the evaluation of the critical exponent ratios
,
, and
are reasonably accurate. Moreover, as can be noticed from the last column of
Table 1, they obey, within the error bars, the hyperscaling relation for all lattice dimensionalities.
4. Concluding Remarks
The Biswas–Chatterjee–Sen model, in its discrete version and defined on three-dimensional Solomon networks, has been studied through extensive Monte Carlo simulations for several values of the local consensus controlling parameter. From the data collapse displayed in panels (d)–(f) of
Figure 1, and from the scaling behavior depicted in
Figure 4,
Figure 5 and
Figure 6, it is clear that the model really undergoes a well-defined second-order phase transition.
It is clear that the present SN sizes do not ensure that the large-size scaling regime of Equations (
9), (10), and (12) has been achieved. In fact, we have seen that finite-size effects are actually present, for instance, in the fourth-order Binder cumulants in
Figure 3. This means that corrections to scaling could be implemented in the scaling relation. This will certainly require extra unknown correction-to-scaling exponents. However, the evaluation of the critical exponents seems to be reasonable, even for the SN sizes employed in
Figure 4,
Figure 5 and
Figure 6, since the data are quite aligned for a proper linear fit (the same in
Figure 3, but only for the largest lattices).
In addition, looking at
Figure 1d,e for the
O and
variables, one can clearly see the finite-size behavior of the data collapse. While for
all sizes collapse to the same curve, for
the collapse systematically tends to the same curve as the SN size increases. For the two largest SNs, namely,
and
, all data are almost superimposed in the whole range of
. This means that we have reached the large-size regime for the universal scaling functions of the variables defined in Equations (
6)–(8). It is also evident from
Figure 1f that the reduced fourth-order Binder cumulant
turns out to be a more robust variable, because there has been no finite-size effects for all SN sizes used herein.
From the critical exponents of the BChS model on SNs, conveyed in
Table 1, it becomes apparent that the model in three dimensions is indeed in a different universality class to the model defined in one and two dimensions. This is an expected result and in agreement with the general trend coming from renormalization group arguments. Here, we also emphasize that our results are different from the three-dimensional BChS model on regular lattices in Ref. [
8]. Despite the critical exponent
being, within the error bars, compatible to the Ising universality class at 3D, this is not true for the other exponents, which could be affected by strong finite-size effects. The present results thus indicate that the BChS model has its own universality class.
It is also interesting to note that although the critical exponents have a sensitive change with the dimension of the network, the critical noise probabilities, responsible for the phase transition in the model, seem not to vary much as the connectivity of the networks is increased by the dimensionality of the lattices.
Finally, an additional study of the BChS model on four-dimensional SN networks should be very interesting from the pure theoretical point of view. Recall that the Ising model has a lower critical dimension
and an upper critical dimension
. The BChS model on SNs has a lower critical dimension
and the present results clearly indicate, if such arguments are valid here, an upper critical dimension greater than
. The question of violation of the hyperscaling relation and a possible unified view of the finite-size scaling valid in all dimensions should also be addressed in this case [
22].