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A dynamical model for competing opinions S.R. Souza1,2∗ and S. Gon¸calves2† 1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Cidade Universita´ria, CP 68528, 21941-972, Rio de Janeiro, Brazil and 2Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CP 15051, 91501-970, Porto Alegre, Brazil (Dated: January 10, 2012) We propose an opinion model based on agents located at the vertices of a regular lattice. Each agent has an independent opinion (among an arbitrary, but fixed, number of choices) and its own degree of conviction. The latter changes every time it interacts with another agent who has a 2 different opinion. The dynamics leads to size distributions of clusters (made up of agents which 1 havethesameopinionandarelocated atcontiguousspatialpositions)whichfollow apowerlaw,as 0 longastherangeoftheinteractionbetweentheagentsisnottooshort,i.e. thesystemself-organizes 2 into a critical state. Short range interactions lead to an exponential cut off in the size distribution n and to spatial correlations which cause agents which have the same opinion to be closely grouped. a Whenthediversityofopinionsisrestrictedtotwo,non-consensusdynamicisobserved,withunequal J population fractions, whereas consensus is reached if the agents are also allowed to interact with 7 those which are located far from them. ] PACSnumbers: 05.65.+b,89.75.-k,87.23.Ge h p - I. INTRODUCTION urally leads to the replacement of old concepts by a new c paradigm, which dominates for a certain period of time, o s Statistical mechanics has turned out to be quite suc- until it is gradually replaced by new ideas. . Inspired by that work, we have developed an opinion s cessful in modeling many systems whose interaction is, c model where agents are placed at the vertices of a reg- in principle, much more complex than those tradition- si ally studied in physics as, in many cases, the systems ular lattice and interact only with those located within y a certain range. Each agent has an opinion and its de- are made up of agents which are endowed with intelli- h gree of conviction. In contrast with many other models, gence and, therefore, the interaction between them de- p such as those developed in Refs. [4, 11, 12], for instance, [ pends on their decisions [1–3]. Nevertheless, simple statistical models have been developed for describing social the interaction between two agents is strictly local, in 1 the sense that it relies only on the agents’ properties, systems[1–4],economy[5–8],etc. Despitethegreatcom- v i.e. their opinions and convictions. Their neighborhood plexityofsuchrealsystems,theirmainpropertiescanbe 2 has no influence on the interaction. The latter primar- reproducedbysimple modelswhichretaintheirunderly- 7 ily affects their degree of conviction. More precisely, the 5 ing features. interactingagents’convictionsareaffectedduringthein- 1 A great deal of effort has been devoted to developing . models for describing the properties of systems made up teraction. If the conviction of one of the agents reaches 1 acertainlowerboundthenitsopinionchangesto thatof 0 ofagentswith competing opinions[1,4,9–14]. This isof the opponent. We therefore take into account the diffi- 2 great relevance as human conflicts very often arise from cultyinpersuadingsomeonewhohasastrongconviction. 1 the simultaneous existence of incompatible opinions in : populations. Different systems,suchashierarchicalsoci- In such a case, it is necessary to change his (her) beliefs v priortotheacceptanceofthenewidea. Furthermore,we i eties [9] or democratic ones [11], where the agents follow X allowthe agentsto interactwith those whichare located theopinionofthelocalmajoritywithinagroup,have,for r instance, been investigated. Most of these models allow beyond their first neighbors. a The dynamics of the opinion distribution in popula- the agents to assume only one of two possible opinions. tions is then studied in the frameworkof this model and Suchspinflipmodels arerepresentativeofmanyrealsit- the remainder of this work is organized as follows. In uations which offer only two possibilities and, therefore, Sec. II we give a detailed description of the model. The are also of great interest [1, 12, 13], besides the similari- results are presented and discussed in Sec. III whereas ties with other physical systems. concluding remarks are drawn at Sec. IV. Theevolutionofscientificparadigmshasbeenrecently modeledinRef.[4]. Theslowdeclineofoldideasandthe quick adoption of new ones are the main characteristics II. THE MODEL ofthe model. Thoseauthorsfindthatthe dynamics nat- The system is built on a regular mesh of N hor- x izontal and N vertical lines, with periodic boundary y ∗Electronicaddress: [email protected] conditions. One agent is placed at each vertex and †Electronicaddress: [email protected] an opinion Oi, among the No possible ones, is ran- 2 domly assigned to the i-th agent, as well as a positive integer C , which corresponds to its degree of convic- i tion. An agent located at vertex (k,l) interacts with any of the neighbors located in vertices (k′,l′) 6= (k,l), where k′ = k − r,k − r + 1,··· ,k + r − 1,k + r and l′ =l−r,l−r+1,··· ,l+r−1,l+r. The ranger is one ofthe model parameters. Thus, eachagenthas 4r(r+1) neighbors. At each step of the dynamics: a) An agent i is sampled with probability proportional to C and one of its neighborsis randomly selected i for interaction, as described below. By doing so weassumethatthe agent’sactivityisrelatedtoits conviction. b) With probability α, another agent is randomly se- FIG. 1: (Color online) Populations of the different opinions lectedamongthe othersandits opinionchangesto as a function of time. Thedifferent groups havebeen shifted anyoftheN possibleones,includingitsown. This o in order to prevent overlap between them. The curves show procedure represents the replacement of the agent the results for r = 1, λ = 1.0, No = 5, α = 1.0×10−6, and by death or substitution due to departure from its Nx =Ny =100. For details, see the text. neighborhood. Its new conviction is selected between 1 and the maximum existing value, in order not to introduce any bias into the system. obtained using λ = 1.0, r = 1, N = 5, α = 1.0×10−6, o and N = N = 100 are shown in Fig. 1. The pop- In step (a) above, nothing is done if the agents have x y ulations have been shifted in order to prevent overlap the same opinion and one then proceeds to step (b). between them. For clarity, we have also restricted the Two agents ‘i’ and ‘j’ may interact only if their opin- time scale to 106, in spite of having carried out simula- ionsaredifferent. Inthiscase,theydoitwithprobability tions up to much larger times, as just mentioned. One sees that no opinion dominates the dynamics. They co- p=exp[−λ(Cmin/Cmax)2], (1) exist in different proportions and one notices that, very often, there is one which is much more popular than the whereCmin (Cmax)is the minimum(maximum)between others. Its dominance lasts for a relatively short time Ci andCj,and λis a parameterwhichis chosenso as to and the popular opinion is replaced by another one. As minimize the interaction between agents for which Ci ≈ a matter of fact, only a small number of opinions are Cj. Thisisbecauseitisveryunlikelythataleaderwould effectively disseminated through the system, the others be influenced by another competing leader (two agents being a small perturbation most of the time. The most with large and similar values of C). On the other hand, popular opinions are replaced by the unimportant ones, ifaleader(largeC value)meetsanordinaryagent(small but few opinions dominate the population at the same C value) who has a different opinion, the latter is very time. likelytobeconvincedbytheformer. Thefunctionalform This conclusion is independent of the number of pos- chosen in this work aims at introducing these features sible opinions N as one sees in Fig. 2, which shows the o into the model. If the agents interact, with probability number of opinions, for which C /(C +C ), the conviction C is increased by one unit i i j i and C decreases by |C −C |. Otherwise, C increases j i j j by one unit and Ci decreases by |Ci −Cj|. If Cj ≤ 0 XNi/Ntotal >ǫ (2) (Ci ≤ 0) then its opinion changes to that of agent i (j) i and C (C ) is set to unit. j i where III. RESULTS AND DISCUSSION No Ntotal =XNi, (3) At the initialization stage, an opinion 1 ≤ O ≤ N i o i=1 andthe convictiondegree1≤C ≤10[25]arerandomly i selectedandassignedtothe i-thagent. The systemthen as a function of time, for ǫ = 0.1 and 0.5. Before carry- evolvesduring,atleast,108steps. Afullstepcorresponds ing out the sum, the indices i are rearranged so that to the time interval during which N ×N intermediate the smallest populations are selected to calculate the x y steps, as explained in (a) and (b) in Sec.II, take place. fractions. The calculations have been carried out for We start by examining the time evolution of the pop- N = 50. It is clear from these results that, although o ulations of groups with the same opinion. The results the composition may change, only very few opinions are 3 actually adopted by the populations. Therefore, the use tiplicity of small clusters should then decrease and their ofafixednumberofopinionsshouldnotbeseenasalim- size distribution would be steeper, as is also seen in Fig. itation of the model as the system naturally eliminates 3. mostof the competing opinions and veryfew ofthem ef- This qualitative reasoning is confirmed by the results fectively take part into the dynamics. We have checked displayedinFig.4,wherethespatialconfigurationofthe that this conclusion still holds if one uses different pa- clustersisshownatrandomlyselectedmoments. Distinct rameters, such as λ=2.0 or r =10. opinions are represented by different gray values (color online). Oneseesthat,compactgroupsareindeedformed for r = 1, whereas the clusters become more and more spatiallydiffuse as r increases. Thus, ourmodelpredicts that long range interactions tend to destroy spatial correlations among opinions, when consensus has not been reached and different opinions coexist in the system. FIG. 2: (Color online) Sum of populations, such that PiNi/Ntotal > ǫ, ǫ = 0.1 and 0.5. The parameter set is the same used in Fig. 1, except for the number of opinions where No =50 is now employed. For details, see thetext. We now turn to the size distribution of clusters made up of neighbors holding the same opinion. The system configuration is analyzed at every 103 full steps and the FIG.3: (Coloronline)Sizedistributionofclustersmadeupof neighboragentswhichhavethesameopinion. Theparameter time average is thus performed. The results are exhib- set corresponds to that used in Fig. 2. The power laws are ited in Fig. 3 for different values of the range r. Two best fit to the results, whose exponents β = 1.65, 1.90, 2.30, agents belong to the same cluster if they have the same and 2.35 are respectively associated with r=1, 2,5, and 10. opinion and their grid coordinates obey (k,l)−(k′,l′)= In the inset, the model predictions for λ=1.0 are compared (±1,0),(0,±1), or (±1,±1). The results clearly show to those for λ=2.0 and r=1 in both cases. For details, see that the size distribution is very sensitive to the inter- thetext. action range r. For cluster sizes up to 10% of the total system, the distribution becomes steeper as r increases, The role played by the parameter λ is illustrated in whereas the development of a big cluster, of approxi- the inset of Fig. 3, where the cluster size distributions mately the size of the total system, becomes more and obtained with λ = 1.0 and λ = 2.0, for r = 1, are com- more pronounced. Since agents interact only if they do pared. The effect on the size distribution is small and is not have the same opinion, the borders between clusters more easily noticed at large sizes where one observes a aretheregionsofstrongactivity. Forshortrangeinterac- slight suppression of big clusters. We have also checked tions,onlyagentswhicharelocatedveryclosetothebor- thattheotherobservablesstudiedinthisworkareweakly dersareallowedtointeract. Therefore,forsmallvaluesof affected if one changes λ in the range 1.0≤λ≤2.0. For r,oneshouldexpecttoobservecompactgroupsofagents, the sake of simplicity, we adopt λ=1.0 from here on. who share the same opinion. This should favor the ap- OnemayalsonoticeinFig.3thatthesizedistribution pearance of medium size clusters. Indeed, large range ofclusters,whosesizesissmallerthan10%ofthesystem values would lead to very diffuse borders and, therefore, size,isfairlyaccuratelyapproximatedbyapowerlaw,i.e. to the disappearance of the coherence among the agents P(s)∝s−β. Theexponentvarieswithrandcorresponds which are close to each other. In the limit of very large to β = 1.65, 1.90, 2.30, and 2.35 for r = 1, 2, 5, and 10, r values,the connectedagentswouldpervadethe system respectively. We have checkedthat the asymptotic value andthedifferentgroupswouldinterpenetrateeachother, is reached for 5<r ≤10. as they would not be compact. This would favorthe ap- These results show that the system self organizes, i.e. pearance of large clusters during the dynamics, whose the configurations are reached without the need of the contribution to the size distribution may also be noticed externaltuningofanyparameter,andthepowerlawsug- inFig.3. Owingtothestrictconservationlaws,themul- geststheexistenceofcriticalbehavior. Theselforganized 4 criticality (SOC) has been discussed in different places tinctscenarios. For smallr, compactclustersareformed [16–20]andhas been observedin manydifferent systems and,occasionally,amalgamateandformaverylargeone. [20–24]. In our case, this suggests that, on rare occa- Duetothelocalityofthecoalescenceprocess(forsmallr sions, consensus would spontaneously be reached. This the interactions take place at the cluster’s borders), this stateshouldsurviveforawhile,untilconflictingopinions happensveryrarelysinceitrequiresstrongspatialcorre- arenucleatedby the noise describedin (b) inSec.II and lations. Then,thisprocessleadstotheexistenceofasize the competition among them would restart and another cutoff. Onthe otherhand,whenlarger valuesareused, opinion would dominate, and so on. the coalescence extends through much larger areas, due to the spreadof the clusters. It therefore makes it easier for correlations to propagate through the entire system. Thus,thepresentmodelpredictstheexistenceofSOCin systems with competing opinions, if the interaction between the agents are not restricted to their contiguous neighborhood. FIG. 5: (Color online) Same as Fig. 3 for different system sizes. The same parameter set has also been employed in the present simulations. The full circles in the up left frame correspond to P(s) ∝ exp(−0.00027s)/s1.65. For details, see FIG. 4: (Color online) Spacial cluster distribution at ran- thetext. domlyselectedmomentsofthedynamics. Agentssharingthe same opinion are depicted by the same gray values (color). Up to this point, we have not investigated the role The simulation has been carried out using No = 5, λ = 1.0, playedbytheparameterα,whichregulatesthefrequency and Nx =Ny =100. For details, see thetext. with which an agent randomly changes his opinion. It contributes with noise, which prevents the system from In order to investigate whether scale invariance is, in freezing when consensus is reached. In this sense, the fact, present in the dynamics, we show, in Fig. 5, the model strongly relies on this parameter to ensure an in- clustersizedistributionsfordifferentrvaluesandsystem teresting dynamics. We have found that it also plays a sizes. The results reveal that, for r = 1, there seems to veryimportantroleindeterminingtheclustersizedistri- be a characteristic scale, since the largestcluster formed bution. We observed that α = 10−5 still leads to power duringthe dynamicsdoes notscalewith the systemsize. law regimes for not too large clusters. However, the in- Infact,asshownbythefullcirclesintheupleftframeof variance of the exponent with the system size shown in thisfigure,thedistributionforr =1andN =N =200 Fig. 5, for r &2, does not hold in this case. This means x y is accurately described by P(s)∝exp(−0.00027s)/s1.65, that,althoughnoiseisneededbythedynamics,toomuch which has a clear exponential cut off. These statements noise destroys the scale invariance, i.e. the agents must are weakenedfor r=2 and are no longer valid for larger keep their opinions for, at least, a short while, in order range values. More precisely, the power law regime ex- to preserve spatial correlations. Since, on the average, tends to larger sizes as the total system size increases N ×N αagentsrandomlychangetheiropinionsateach x y for r ≥ 2. Actually, there is a range value, between 2 step, one sees that there is no unique value of α that and 5, for which the power law regime is an adequate would ensure scale invariance for arbitrary system sizes description of the size distribution, except for very large sinceonemayalwaysfindasizeforwhichtoomuchnoise clusters, since finite size effects have to be considered is added to the system at each time step, destroying the for those clusters. Thus, the dynamics leads to two dis- spatial correlations. This shortcoming may be avoided 5 by redefining α as the total rate per step, i.e. propor- ionsareallowed. Themodelingofsuchsystemsisofgreat tional to (N N )−1, so that the desired amount of noise interest and have been extensively studied [1, 10, 12–14] x y is introduced into the system during the dynamics, for since there are many situations in real life where binary any system size. choices have to be made [1]. Furthermore, the above results,associatedwithFig.2, alsosuggestthatthis sce- nario should retain most of the properties of real systems. The results of the model simulation obtained for N = N = 100 are displayed in Fig. 7, which exhibits x y the difference between the populations with opinion 1 (n1)andopinion2(n2). AsinRefs.[12,13],whereanon- consensusopinionmodelhasbeenproposedandstudied, our model allows for the dynamic coexistence of the two conflicting opinions with unequal population fractions. This is seen in the left panels of this figure which show the results for short range interactions between neighbors, i.e. r = 1 and 2. This is in agreement with Ref. [12] which considers the interaction between the closest neighbors. For larger r values, one sees that consensus is reached and it lasts for a long period. We have fol- lowed the dynamics during much longer time scales and confirmed this feature. Owing to the noise introduced FIG.6: (Coloronline)Timeaveragedconvictiondistribution. by the random change of the agents’ opinions, regulated The parameter set is the same employed in the calculations shown in Fig. 3. For details, see thetext. by the parameter α, the status quo does not last forever and, after being the overwhelmingly dominating opinion Since the agents’ convictions play an important role for a long period, its replacement occurs very quickly in the dynamics, as it directly influences their resistance andthe otheropinionbecomes the consensus,andsoon. to the adoption of new paradigms, we also examine this Therefore,ourmodelpredictsthatatransitionfromnon- quantity. Thus, Fig. 6 displays the time averaged con- consensus to consensus occurs as the interaction among viction distribution for different values of r. As in real the agents changes from short to long range. life, most of the agents have a low degree of conviction andthe systemhasveryfewleaders(largeC values),i.e. the distributiondecaysexponentially. As expected, for a IV. CONCLUDING REMARKS given value of C, the distribution falls off as r increases, since longer range interaction allows the agents to en- We have developed a model for the dynamics of com- counter others with different opinions more often (as is peting opinions, which is based on the agents’ degree of illustrated in Fig. 4, agents with the same opinion tend conviction and on the range of the interaction between to be closer for small r values and they do not interact). them. It predicts that, even when many different opinions are allowed, only very few of them are really in use by the agents during the dynamics. This is, in fact, observed in real life when, for instance, at the beginning of an election process, many candidates running for a political office start with not too different opportunities but, after a while, very few dominate the voters’ pref- erences. The model also predicts that the size distribution of clusters, made up of agents which are located in contiguous spatial positions and share the same opinion, follows a power law. That distribution is reached independently of the initial conditions, i. e. the dynamics leads to SOC [16–20], as long as the interaction between the agents is not restricted to too close neighbors. When only two opinions are allowed, the model leads to non-consensusdynamics,whichqualitativelyagreeswith the non-consensus model proposed in Ref. [12]. On the FIG. 7: (Color online) Time evolution of the difference between populations with opinion 1 and 2. Except for No =2, other hand, if the agents also interact with those who the parameter set is the same employed in the calculations arelocatedrelativelyfarfromthem, consensusisquickly shown in Fig. 4. For details, see thetext. reached and it lasts for a long time. The dominating opinion is occasionally replaced, but there is consensus We finally examine the dynamics when only two opin- almostallthetime. Ourmodelthenprovidesameansto 6 simulatemanyofthepropertiesofrealsystemsbychang- initiatives of CNPq/FAPERJ under Contract No. 26- ing a parameter which has a very simple interpretation 111.443/2010 and CNPq/FAPERGS , for partial finan- on physical systems, i. e. the range of the interaction cial support. between the agents. It also contrast with other models asthe interactionbetweenagentsaffects theirconviction in first place and their opinions change only after their paradigms have been corroded. 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