Papers in Physics, vol. 6, art. 060003 (2014) Received: 11 March 2014, Accepted: 1 August 2014 Edited by: G. Martinez Mekler Reviewed by: F. Bagnoli, Dipartimento di Fisica ed Astronomia, Universita degli Studi di Firenze, Italy Licence: Creative Commons Attribution 3.0 DOI: http://dx.doi.org/10.4279/PIP.060003 www.papersinphysics.org ISSN 1852-4249 Critical phenomena in the spreading of opinion consensus and disagreement A. Chacoma,1 D. H. Zanette1, 2∗ We consider a class of models of opinion formation where the dissemination of individual opinions occurs through the spreading of local consensus and disagreement. We study the emergence of full collective consensus or maximal disagreement in one- and two- dimensional arrays. In both cases, the probability of reaching full consensus exhibits well-defined scaling properties as a function of the system size. Two-dimensional systems, in particular, possess nontrivial exponents and critical points. The dynamical rules of our models, which emphasize the interaction between small groups of agents, should be considered as complementary to the imitation mechanisms of traditional opinion dynamics. I. Introduction The remarkable regularities observed in many hu- man social phenomena —which, in spite of the disparate behavior of individual human beings, emerge as a consequence of their interactions— have since long attracted the attention of physicists and applied mathematicians. Collective manifes- tations of human behavior have been mathemat- ically modeled in a variety of socioeconomic pro- cesses, such as opinion formation, decision making, resource allocation, cultural and linguistic evolu- tion, among many others, often using the tools pro- vided by statistical physics [1]. The stylized nature of these models emphasizes the identification of the generic mechanisms at work in human interactions, as well as the detection of broadly significant fea- ∗E-mail: zanette@cab.cnea.gov.ar 1 Instituto Balseiro and Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Ŕıo Negro, Argentina. 2 Consejo Nacional de Investigaciones Cient́ıficas y Técnicas, Argentina. tures in their macroscopic outcomes. They provide the key to a deep insight into the common elements that underlie those processes. Models of opinion formation constitute a central paradigm in the mathematical description of social processes from the viewpoint of statistical physics. Starting in the seventies and eighties [2–5], much work —which we cannot aim at inventorying here, but which has been comprehensibly reviewed in re- cent literature [1]— has exploited the formal re- semblance between opinion spreading and spin dy- namics in order to apply well-developed statistical techniques to the analysis of such models. The key mechanism driving most agent-based models of opinion formation is imitation. For in- stance, in the voter model —to which we refer sev- eral times in the present paper— the basic interac- tion event consists in an agent copying the opinion of another agent chosen at random from a speci- fied neighborhood. At any given time, the opin- ion of each agent adopts one of two values, typi- cally denoted as ±1. The voter model can be ex- actly solved for populations of agents distributed over regular (hyper)cubic arrays in any dimension 060003-1 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. [6]. For infinitely large populations, it is character- ized by the conservation of the average opinion. In one dimension, a finite population always reaches an absorbing state of full collective consensus, all agents sharing the same opinion. The probability of final consensus on either opinion coincides with the initial fraction of agents with that opinion, and the time needed to reach the absorbing state is of the order of the population size squared [1]. In this paper, we present an introductory anal- ysis of a class of models where opinion dynamics is driven by the spreading of consensus and dis- agreement, rather than by the dissemination of in- dividual opinions. The basic concept behind these models is that agreement of individual opinions in a localized portion of the population may promote the emergence of consensus in the neighborhood while, in contrast, local disagreement may inhibit the growth of, or even decrease, the degree of con- sensus in the surrounding region. In real social systems, the mechanism of consensus and disagree- ment spreading should be complementary to the direct transmission of opinions between individual agents. In our models, however, we disregard the latter to focus on the dynamical effects of the for- mer. Since the degree of consensus can only be defined for two or more agents, the spreading of consensus and disagreement engages groups of agents rather than individuals. Such groups are, thus, the ele- mentary entities involved in the social interactions [7–11]. We stress that several other social phenom- ena — related, notably, to decision making [10] and resource allocation [12]— are also based on group interactions that cannot be reduced to two-agent events. In the class of models analyzed here, each interaction event is conceived to occur between two groups: an active group G and a reference group G′. As a result of the interaction, the agents in G change their individual opinions in such a way that the level of consensus in G approaches that of G′. This generic mechanism extends dynamical rules where the opinion of each single agent changes in response to the collective state of a reference group [1, 8, 13, 14]. The size and internal structure of the interacting groups, as well as the precise way in which opinions are modified in the active group with respect to the reference group, defines each model in this class. For the sake of concrete- ness, we limit the analysis to systems where, as in the voter model, individual opinions can adopt two values (±1). In the next section, we analyze the case where both the active group and the reference group are formed by two agents, and the popula- tion is structured as a one-dimensional array. In this case, the system admits stationary absorbing states of full consensus and maximal disagreement, with simple scaling laws with the population size. In Section III., we study a two-dimensional version of the same kind of model with larger groups, where nontrivial critical phenomena —not present in the one-dimensional case— emerge. Results and per- spectives are summarized in the final section. II. Two-agent groups on one- dimensional arrays We begin by considering the simple situation where each of the two groups involved in each interaction event is formed by just two agents. The situation within each group, thus, is one of either full con- sensus (when the two agents bear the same opinion, either +1 or −1) or full disagreement (when their opinions are different). We take a population where agents are distributed on a one-dimensional array, and consecutively labeled from 1 to N. Periodic boundary conditions are applied at the ends. At each time step, we choose four contiguous agents, say, i−1 to i+2. The central pair i, i+1 acts as the reference group G′. If they are in disagreement, the agents i−1 and i+2 respectively adopt the opinions opposite to those of i and i+ 1 with probability pD, while with the complementary probability 1 − pD nothing happens. If, on the other hand, i and i + 1 agree with each other, i − 1 and i + 2 copy the common opinion in G′ with probability pC , while with probability 1 − pC nothing happens. In this way, both consensus and disagreement spread from G′ outwards, to the left and right. The probabili- ties pC and pD control the relative frequency with which consensus and disagreement are effectively transmitted. The left panel of Fig. 1 illustrates the states of the four consecutive agents in the two possible outcomes of the interaction (up to opinion inversions). It is not difficult to realize that, for pD = pC = 1, our one-dimensional array is equivalent to two in- tercalated subpopulations —respectively occupying even and odd sites— each of them evolving accord- 060003-2 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. 0 200 400 600 800 1000 1200 1400 1600 0 50 100 150 200 time G' G Figure 1: Left: The two possible outcomes of the interaction, up to opinion inversions, for four con- secutive agents along the one-dimensional array. The active and the reference groups, G and G′, are respectively formed by the outermost and inner- most agents. Right: Time evolution of a 200-agent array with n+(0) = 0.5 and pD = pC = 1. Black and white dots correspond, respectively, to opin- ions +1 and −1. At time t = 1534, an absorbing state of maximal disagreement is reached. ing to the voter model. The dynamical rules are reduced in this case to binary interactions between agents. In fact, whatever the opinions in group G′ at each interaction event, agent i − 1 and i + 2 re- spectively copy the opinions of i + 1 and i. Now, since the voter model always leads a finite popu- lation to an absorbing state of full consensus, the final state of our system can be one of full consen- sus on either opinion, or a state of maximal dis- agreement where opposite opinions alternate over the sites of the one-dimensional array. In the lat- ter, the two neighbors of each agent with opinion +1 have opinion −1 and vice versa. The right panel of Fig. 1 shows the evolution of a 200-agent array for n+(0) = 0.5 and pD = pC = 1, black and white dots respectively corresponding to opinions +1 and −1. At any given time, the population is divided into well-defined domains either of consensus in one of the opinions or disagreement. Note that the do- main boundaries show the typical diffusive motion found in stochastic coarsening processes [1, 15]. Taking into account that, in the voter model, the probability of ending with full consensus on opin- ion +1 is given by the initial fraction of agents with that opinion, n+(0), and assuming that the ini- tial distribution of opinions is homogeneous over the array, the probability that our system ends in a state of full consensus on either opinion is Pcons = n 2 +(0)+n 2 −(0) = 1−2n+(0)+2n2+(0). Note 0.4 0.6 0.8 1.0 N = 130 N = 230 N = 430 N = 930 P c o n s 0.0 0.1 0.2 0.3 0.4 0.5 10 -3 10 -2 N − 2 T n + (0) Figure 2: Numerical results for consensus and dis- agreement spreading on a one-dimensional array with pD = pC = 1, obtained from 10 3 realizations for each parameter set (see text for details). Upper panel: Probability of reaching full consensus, Pcons, as a function of the initial fraction of agents with opinion +1, n+(0), for four values of the popula- tion size N. Lower panel: Total time T needed to reach the final absorbing state, normalized by the squared population size N2. Since both Pcons and N−2T are symmetric with respect to n+(0) = 1/2, only the lower half of the horizontal axis is shown. that this coincides with the probability that, in the initial state, any two contiguous agents are in con- sensus. Moreover, we know that the time needed to reach an absorbing state in the one-dimensional voter model is proportional to N2, a result that should also hold in our case. The upper panel of Fig. 2 shows numerical results for the probability of final full consensus Pcons, de- termined as the fraction of realizations that ended in full consensus out of 103 runs, as a function of n+(0) and for several population sizes N. The curve is the analytic prediction given above. The result is analogous to the probability of final con- sensus found in Sznajd-type models [13]. The lower panel shows the total time T needed to reach the fi- nal absorbing state (of either consensus or disagree- 060003-3 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0.5 0.6 0.7 0.8 0.9 1.0 N = 130 N = 230 N = 430 N = 930 P c o n s p D 300 600 900 0.01 0.1 w id th N Figure 3: Probability of reaching full consensus, Pcons, as a function of the probability pD, with pC = 1 and for four values of the system size N. Results were obtained averaging over 103 realiza- tions for each parameter set. Insert: Width of the variation range of Pcons as a function of N. The straight line has slope −1. ment), averaged over 103 realizations and normal- ized by N2. As expected, both Pcons and N −2T are independent of the population size. When pD 6= pC , the two intercalated subpopu- lations cannot be considered independent of each other any more. If pD < pC , for instance, an opin- ion prevailing in one of the subpopulations will in- vade the other subpopulation faster than the op- posite opinion, thus favoring the establishment of collective consensus. To analyze this asymmetric situation, we first fix pC = 1 and let pD vary in (0, 1), so that the spreading of consensus is more probable than that of disagreement. The main plot in Fig. 3 shows numerical results for Pcons, measured as explained above, as a function of pD and for four values of N. In all the realizations, n+(0) = 0.5, and the two opinions are homoge- neously distributed over the population. As pD decreases below 1, the probability of reaching full consensus grows rapidly, approaching Pcons = 1. As N grows, moreover, the change in Pcons is more abrupt. Fitting of a sigmoidal function to the data of Pcons vs. pD near pD = 1 makes it possible to assign a width to the range where Pcons changes between 1 and 0.5. The insert of Fig. 3 shows this width as a function of the system size N in a log-log plot. The slope of the linear fitting is −1.00±0.02. 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 N = 130 N = 230 N = 430 N = 930 varying p C P c o n s N(1−p D ), N(1−p C ) varying p D Figure 4: Probability of reaching full consensus, Pcons, as a function of N(1−pD) when varying pD with pC = 1, and as a function of N(1 −pC ) when varying pC with pD = 1. Therefore, the width is inversely proportional to N. The facts that Pcons = 0.5 for pD = 1 and for all N, and that the width of the range where Pcons changes decreases as N −1, make it possible to conjecture the existence of a function Φ(u), with Φ(0) = 0.5 and Φ(u) → 1 for large u, such that Pcons = Φ[N(1 − pD)]. To test this hypothesis, we have plotted our numerical data for Pcons against N(1−pD) in Fig. 4. The results are those in the up- per half of the plot (“varying pD”). The collapse of the data for different N on the same curve confirms the conjecture. Analogous results were obtained when fixing pD = 1 and pC was varied. Now, Pcons drops to 0 in a narrow interval for pC just below 1, indi- cating the prevalence of disagreement. Again, the width of the interval is proportional to N−1. The results in the lower half of Fig. 4 (“varying pC ”) illustrate the collapse of the corresponding values of Pcons when plotted against N(1 −pC ). In our numerical realizations with pD 6= pC , we have also recorded the average time T needed to reach the final absorbing state. Figure 5 shows re- sults for N−2T in the case where pC = 1 and pD changes (cf. lower panel of Fig. 2). In contrast with the case with pD = pC = 1, rescaling of the time T with N2 leaves a remnant discrepancy between re- sults for different population sizes N. Specifically, for pD < 1, T grows faster than N 2. Moreover, T 060003-4 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. 0.5 0.6 0.7 0.8 0.9 1.0 0.01 0.02 0.03 0.04 N = 130 N = 230 N = 430 N = 930 N − 2 T p D Figure 5: Total time T needed to reach the final absorbing state, normalized by the squared popu- lation size N2, as a function of the probability pD (pC = 1). Bézier curves have been plotted as a guide to the eye. is nonmonotonic as a function of pD, exhibiting a minimum which shifts towards pD = 1 as N grows. The same dependence with N and pC is observed when we fix pD = 1 and let pC vary. Summarizing our results for a one-dimensional population with two-agent groups, we can say that the possibility that both consensus and disagree- ment spread over the system makes it possible to find absorbing collective states of either full con- sensus, with all the agents having the same opin- ion, or maximal disagreement, where opposite opin- ions alternate between consecutive neighbor agents. For large populations, the relative prevalence of collective consensus and disagreement is controlled by how the probabilities pD and pC compare with each other. Our results suggest that, in the limit N → ∞, the condition pC > pD univocally leads to full consensus and vice versa. For smaller sizes, however, the system can approach full consensus even when pD > pC , and vice versa —presumably due to finite-size fluctuations. III. Larger groups on two- dimensional arrays A two-dimensional version of the above model, where agents occupy the N = L × L sites of a regular square lattice with periodic boundary con- ditions, can be defined as follows. The reference group G′ at each interaction event is a randomly chosen 2×2-agent block. The corresponding active group G is formed by the eight nearest neighbors to the agents in G′ which are not in turn mem- bers of the reference group. The active group, thus, surrounds G′. Of the sixteen possible opinion con- figurations of the reference group, two correspond to full consensus —with the four agents sharing the same opinion— and six correspond to maximal dis- agreement —with two agents in each opinion. The remaining eight configurations correspond to par- tial consensus, with only one agent disagreeing with the other three. The dynamical rules are the fol- lowing: (1) if G′ is in full consensus, all the agents in G copy the common opinion in G′; (2) if G′ is in maximal disagreement, each agent in G adopts the opinion opposite to that of the nearest neigh- bor in G′; (3) otherwise, nothing happens. Hence, both consensus and disagreement spread outwards from the reference group. Probabilities pD and pC for the spreading of disagreement and consensus are introduced exactly as above. The left part of Fig. 6 shows, up to rotations and opinion inversions, the three possible outcomes of a single interaction event. The states of full collective consensus —with all the agents in the population having the same opinion— and of maximal collective disagreement —with the two opinions alternating site by site along each direction over the lattice— are ab- sorbing states, in correspondence with the one- dimensional case. However, for pD = pC = 1, the system cannot be reduced anymore to a collection of sublattices governed by the voter model. The definition of G and G′ establish now correlations between the opinion changes in the active group at each interaction event. Moreover, some opinion configurations in the reference group induce evolu- tion in the active group, while others do not. Figure 6 shows, to its right, four snapshots of a 120×120- agent population, along a realization starting with n+(0) = 0.35 and pD = pC = 1. Note the for- mation of consensus clusters at rather early stages, and the final prevalence of disagreement. The line boundaries between disagreement regions are also worth noticing. Following the same lines as for the one- dimensional array, we study first the probability Pcons of reaching full collective consensus as a func- 060003-5 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. t = 3207 G t = 12 G' t = 1017 t = 0 Figure 6: Left: The three possible outcomes of the interaction, up to ±90◦ rotations and opinion in- versions, on the two-dimensional lattice. The ac- tive and the reference groups, G and G′, are re- spectively formed by the outermost and innermost agents. Right: Four snapshots of a population with L = 120, for n+(0) = 0.35 and pD = pC = 1, in- cluding the initial condition and two intermediate states. At time t = 3207, an absorbing state of maximal disagreement has been reached. Black and white dots correspond, respectively, to opinions +1 and −1. tion of the initial fraction of agents with opinion +1, n+(0), in the case pD = pC = 1. Opinions are homogeneously distributed all over the popu- lation. For very small n+(0), as expected, we find Pcons ≈ 1. However, in sharp contrast with the one- dimensional case (see Fig. 3), Pcons remains close to its maximal value until n+(0) ≈ 0.35, where it drops abruptly to Pcons ≈ 0. The width of the tran- sition zone decreases as a nontrivial power of the system size, ∼ L0.83±0.04, as illustrated in the insert of Fig. 7. Our best estimate for the critical value of n+(0) at which Pcons drops is n crit + = 0.353±0.001. The main plot in the figure shows the collapse of numerical measurements of Pcons as a function of n+(0) for different sizes L, averaged over 100 real- izations, when plotted against the rescaled shifted variable L0.83[n+(0) − 0.353]. These results suggest that, for very large pop- ulations, the probability of reaching full consen- sus jumps discontinuously from Pcons = 1 to 0 at −4 −2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 L = 90 L = 120 L = 150 L = 180 L = 210 P c o n s L 0.83 [n + (0) − 0.353] 10 100 0.01 0.1 w id th L Figure 7: Numerical results for the probability of reaching full consensus, Pcons, on a two-dimensional lattice with pD = pC = 1, obtained from 100 real- izations for each parameter set. Collapse for several system sizes L is obtained plotting Pcons against L0.83[n+(0) − 0.353]. Insert: Scaling of the width of the transition zone of Pcons, determined from fit- ting a sigmoidal function, as a function of the size L. The straight line has slope −0.83. n+(0) = n crit + . Compare this with the smooth, size- independent behavior of the one-dimensional case. Note also that ncrit+ is close to, but does not coin- cide with, n+(0) = 1/3. At this latter value, in the initial condition with homogeneously distributed opinions, the probability of finding a 2 × 2-agent block in full consensus becomes lower than that of maximal disagreement as n+(0) grows. In the above simulations, we have also measured the average total time T needed to reach the fi- nal absorbing state. Results are shown in Fig. 8. Again in contrast with the one-dimensional case, T exhibits a remarkable change in its scaling with the system size as n+(0) overcomes the critical value ncrit+ . Going now to the dependence of Pcons on the probability of disagreement spreading pD —with pC = 1 and n+(0) = 0.5— it qualitatively mir- rors that of the one-dimensional case, shown in Fig. 3. Namely, as pD decreases from 1, Pcons grows from 0 to 1 in an interval whose width decreases with the population size. In the two- dimensional system, however, the transition takes place at a critical probability pcritD that can be 060003-6 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. 0.20 0.25 0.30 0.35 0.40 10 100 1000 10000 L = 90 L = 120 L = 150 L = 180 L = 210 T n + (0) Figure 8: Total time T needed to reach the final absorbing state in a two-dimensional lattice, as a function of n+(0), for different sizes L. clearly discerned from pD = 1. Our estimate is pcritD = 0.984 ± 0.002. Moreover, the scaling of the transition width with the population size exhibits a nontrivial exponent, decreasing as L−0.93±0.05. Collapse of the rescaled numerical results for var- ious sizes, obtained from averages of 100 realiza- tions, are shown in Fig. 9, where we plot Pcons as a function of L0.93(0.984−pD) (cf. Fig. 4). The insert displays the power-law dependence of the width on the size L. Analogous results are obtained if the probability of consensus spreading pC is varied, with pD = 1. Finally, we have found that the transition in Pcons as a function of the disagreement probability pD shows a dependence on the initial fraction of agents with opinion +1. To characterize this effect in a way that highlights the relative prevalence of disagreement and consensus, we have measured the value of pD at which the probability of getting full collective consensus reaches Pcons = 0.5, as a func- tion of n+(0). The parameter plane (n+(0), pD), thus, becomes divided into regions where a final state of full consensus is more probable than that of maximal disagreement, and vice versa. Results for a 120 × 120-agent population are presented in Fig. 10. In summary, while spreading of consensus and disagreement on a two-dimensional lattice bears superficial qualitative similarity with the one- dimensional case, the probability that the popula- tion reaches full collective consensus in two dimen- −2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 L = 90 L = 120 L = 150 L = 180 L = 210 P c o n s L 0.93 (0.984 − p D ) 10 100 0.01 0.1 w id th L Figure 9: Collapse of numerical results for the prob- ability of reaching full consensus, Pcons, on a two- dimensional lattice with pC = 1 and n+(0) = 0.5, for several system sizes L when plotted against L0.93(0.984 − pD). Insert: Scaling of the width of the transition zone of Pcons as a function of the size L. The straight line has slope −0.93. 0.35 0.40 0.45 0.50 0.96 0.97 0.98 0.99 1.00 consensus p D n + (0) disagreement Figure 10: Zones of relative prevalence of full consensus and maximal disagreement in a two- dimensional lattice with L = 120, plotted on the parameter plane (n+(0), pD). Symbols stand for numerical results, and the curve serves as a guide to the eye. 060003-7 Papers in Physics, vol. 6, art. 060003 (2014) / A. Chacoma et al. sions exhibits a quite different dependence on the system size, on the initial conditions, and on the spreading probabilities. In particular, our results reveal the existence of critical phenomena involv- ing scaling laws with nontrivial exponents. IV. Conclusion In this paper, we have considered the emergence of collective opinion in a population of interacting agents where, instead of imitation between individ- ual agents, opinions are transmitted through the spreading of local consensus and disagreement to- ward their neighborhoods. The basic interacting units in this mechanism are not individual agents but rather small groups of agents which mutually compare their internal degrees of consensus and modify their opinions accordingly. In this sense, it extends the basic mechanism underlying such mod- els as the majority-rule and Sznajd-like dynam- ics [1, 8, 13], where the opinion of each individual agent changes in response to the collective state of a reference group. It is expected that in real so- cial systems the dissemination of individual opin- ions through agent-to-agent imitation on one side, and the spreading of consensus and disagreement by group interaction on the other, are complemen- tary mechanisms simultaneously shaping the over- all opinion distribution. Here, in order to gain in- sight on the specific effects of the second class, we have focused on models solely driven by the spread- ing of consensus and disagreement. The combined effects of the two mechanisms is a problem open to future work. Our numerical simulations concentrated on two-opinion models evolving on one- and two- dimensional arrays [14]. In both cases, absorbing states with all the population bearing the same opinion (full consensus) and with half of the popu- lation in each opinion (maximal disagreement) are possible final states for the system. Maximal dis- agreement states are characterized by alternating opinions between neighbor sites along the arrays. A relevant quantity to characterize the behav- ior is the probability of reaching full consensus, as a function of the initial condition —i.e., the ini- tial fraction of the population with each opinion— and of the relative probabilities of consensus and disagreement spreading. The total time needed to reach the final absorbing state, averaged over re- alizations, has also been measured as a character- ization of the dynamics. We have found that, in several cases, these quantities display critical phe- nomena when the control parameters are changed, with power-law scaling laws as functions of the sys- tem size, pointing to the presence of discontinuities in the limit of infinitely large populations. It is in- teresting to remark that the scaling laws are rather simple for one-dimensional arrays, but involve non- trivial exponents and critical points in the case of two-dimensional systems. Within the same one- and two-dimensional mod- els analyzed here, an aspect that deserves further exploration is the dynamics and mutual interaction of the opinion domains that develop since the first stages of evolution (Figs. 1 and 6). However, the most interesting extension of the present analysis should progress along the direction of considering more complex social structures. The interplay be- tween the dynamical rules of consensus and dis- agreement spreading and the topology of the in- teraction pattern underlying the population might bring about the emergence of new kinds of collec- tive self-organization phenomena. Acknowledgements - We acknowledge enlighten- ing discussions with Eduardo Jagla. Financial sup- port from ANPCyT (PICT2011-545) and SECTyP UNCuyo (Project 06/C403), Argentina, is grate- fully acknowledged. [1] C Castellano, S Fortunato, V Loreto, Statis- tical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009). [2] W Weidlich, The statistical description of po- larization phenomena in society, Br. J. Math. Stat. Psychol. 24, 251 (1971). [3] R Holley, T Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model, Ann. Probab. 3, 643 (1975). [4] S Galam, Y Gefen, Y Shapir, A mean behavior model for the process of strike, J. Math. Sociol. 9, 1 (1982). 060003-8 Papers in Physics, vol. 6, art. 060003 (2014) / A. 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