Phase change in an opinion-dynamics model with separation of time scales
Gerardo Iñiguez, János Kertész, Kimmo K. Kaski, R. A. Barrio
PPhase change in an opinion-dynamics model with separation of time scales
Gerardo I˜niguez , J´anos Kert´esz , , Kimmo K. Kaski , and R. A. Barrio , BECS, School of Science and Technology, Aalto University, P.O. Box 12200, FI-00076, Institute of Physics and HAS-BME Cond. Mat. Group, BME, Budapest, Budafoki ´ut 8., H-1111, and Instituto de F´ısica, UNAM, Ciudad Universitaria, C.P. 04510, M´exico D.F. (Dated: November 2, 2018)We define an opinion formation model of agents in a 1d ring, where the opinion of an agent evolvesdue to its interactions with close neighbors and due to its either positive or negative attitude towardthe overall mood of all the other agents. While the dynamics of the agent’s opinion is describedwith an appropriate differential equation, from time to time pairs of agents are allowed to changetheir locations to improve the homogeneity of opinion (or comfort feeling) with respect to their shortrange environment. In this way the time scale of transaction dynamics and that of environmentupdate are well separated and controlled by a single parameter. By varying this parameter wediscovered a phase change in the number of undecided individuals. This phenomenon arises fromthe fact that too frequent location exchanges among agents result in frustration in their opinionformation. Our mean field analysis supports this picture.
PACS numbers: 89.75.Fb, 87.23.Ge, 64.60.aq
I. INTRODUCTION
How are opinions formed? In sociology this is one ofthe basic questions, but it is also highly relevant for poli-tics, innovation spreading, decision making, and the gen-eral well feeling of people [1–3]. This complex process de-pends on various factors or components like confidence,attitudes, communities or media effects [4]. Recently,much effort has been invested in modeling different as-pects of opinion dynamics and these models are in manyways related to those of physics [5, 6]. Unfortunately,the empirical observations are rather sparse. Therefore,the usual strategy is to concentrate on some particularfeatures by making plausible assumptions for a model,and comparing its results with expectations. Here wewill follow this line of study.Our starting point is that the comfort feeling of anindividual depends on his/her embedding in the soci-ety. We get friends mostly with people who are simi-lar to us, share our opinions, tastes etc. In sociologythis is called homophily and is known to be the majorgoverning principle in friendship formation [1]. In termsof physics, this corresponds to ferromagnetic interactions[7]. In the language of opinion dynamics this means that:a) The opinion of an individual gets adjusted to that ofhis/her friendship neighborhood; b) An individual seeksthe neighborhood of alike others. Here a) has been thebasis of many opinion-dynamics models, both discreteand continuous [8–14]. On the other hand b) has beeninvestigated in the framework of coevolving networks [15–21], where the connections between individuals are notthere forever but can be changed in parallel with theevolution of the opinions in order to increase the level ofsatisfaction in the system [22–28].Recently we have introduced a coevolving networkmodel [28, 29], where not only short range ferromagneticinteractions but also long range interactions were takeninto account. This corresponds to the fact that, although our opinion is strongly influenced by our close friends, weare not independent of the general mood of the society.However, the impact of the society as a whole does nothave to be ferromagnetic. As known from sociology again[1], all individuals have two kinds of driving forces withrespect to the society: We want to be similar to the aver-age around us to use the society’s protecting power and,at the same time, we want to be different to be distin-guished as individuals. For every individual these con-flicting components are present in different proportions,resulting in either net positive or net negative attitudewith respect to the overall opinion of other individuals.This effect was taken into account [28] by an attitudeparameter α considered fixed or quenched to each indi-vidual. Since the attitude parameter can have positiveor negative sign, it constitutes a source of frustration [30]in the system.In [28, 29] we also introduced a separation of timescales for different opinion formation mechanisms. Whilecommunications go on all the time leading to a quasi-continuous adjustment of the individuals’ opinions, ittakes more effort to make new friends than to quit withold ones. Therefore, we introduced a measure of timeseparation, which characterizes this difference by allow-ing for changes in the network neighborhoods after g timesteps of the difference equation governing the opinion up-date. We found interesting effects as a function of g andthe attitude parameter α : For small values of g , wherethe rewiring process is very rapid and only two commu-nities eventually develop, the attitude parameter plays aminor role and the α distributions in them were foundto be broad and similar. However, for the intermediatevalues of g the smaller communities have a rather narrowdistribution with mostly negative α values, while the dis-tributions for larger communities are broad and shiftedtoward positive α values. Naturally, the agents with neg-ative α ’s do not feel comfortable in a large homogeneouscommunity, thus they tend to build smaller ones. a r X i v : . [ phy s i c s . s o c - ph ] N ov The aim of the present paper is to understand bet-ter the role of the attitude parameter and the separationof time scales in the coevolution of opinion and networkstructure of the underlying system. In order to do so,we define a model on a ring, and keep this topology pre-served. Therefore, instead of rewiring the network weallow for location exchanges between agents by carry-ing their individual opinions and attitudes. This corre-sponds to a situation where the agent looks for a betterenvironment to live in, and is reminiscent of Schelling’scheckerboard model for residential segregation, where therelocation of agents with a mild preference for having afew alike neighbors in a static lattice can lead to fullysegregated outcomes [31–37]. The decision whether suchan exchange is made is assumed dependent only on theshort range interactions. However, in the opinion forma-tion the attitude toward the social mood plays also a role.As the possibilities for finding new environments are lim-ited, an amount of frustration will remain in the systemfor not too large values of the time separation parameter g . Interestingly, we see as a function of g a rather sharp,phase transition-like change to a state without frustra-tion as the individuals get enough time to form a firmopinion.This paper is organized as follows. In Section II we in-troduce the model in detail. In Section III we present thenumerical results. In Section IV a mean field calculationis presented, giving account for the variation in the num-ber of undecided agents. Finally we draw conclusions. II. MODEL
As in [28, 29], we study the dynamics of opinion for-mation in a network with a fixed number of individu-als or agents ( N ) to whom a simple question is posed.For the network connectivity between agents, we hereassume a 1d ring topology instead of a more complexnetwork topology we studied earlier. A state variable x i ∈ [ − x lim , x lim ] (for fixed x lim >
0) is associated witheach individual i , which measures the agent’s instanta-neous inclination concerning the question at hand, whilethe network links represent the presence of discussionsbetween agents related to this question. The time scalefor discussions or exchange of information between indi-viduals (“transactions”) is dt , while the time scale for ageneralized change of connections in the network (“gener-ation”) is T . These two quantities are related by T = gdt ,where the parameter g defines the number of transactionsper generation.The dynamics of the agent’s state variable x i can bewritten as ∂x i ∂t = f s ( { x j } s ) x i + f l ( { x j } l ) α i , (1)where the random parameter α i ∈ [ − α lim , α lim ] (for fixed α lim >
0) accounts for the agent’s own attitude towardsoverall or public opinion. The short range interaction term f s ( { x j } s ) x i describes the direct influence over i ofthe subset of ‘close’ agents { x j } s , while the long rangeinteraction term f l ( { x j } l ) α i measures the indirect effectof the subset of ‘far’ agents { x j } l modulated by the at-titude of i . The system consist of a ring (a chain withperiodic boundary conditions) where the short range in-teractions take place over the first m neighbors of eachagent, so the number of short range connections is 2 m .The long range interaction takes into account the averageof opinion over the rest of agents in the network, that is, f s ( { x j } s ) x i = (cid:104) x (cid:105) ( m ) i sgn( x i ) x i = | x i | m m (cid:88) (cid:96) =1 x i ± (cid:96) , (2) f l ( { x j } l ) α i = (cid:104) x (cid:105) ( N − m ) i α i = α i N − − m [ N/ (cid:88) (cid:96) = m +1 x i ± (cid:96) , (3)where sgn( x i ) denotes the sign of x i . Observe that m < ( N − /
2. Once the opinion of an agent reaches anyof the limit values ± x lim , it stays fixed for the rest ofthe dynamics and the agent is said to be decided . This isbecause we attempt to describe a state of total convictionthat is unlikely to change anymore, like in a ballotingprocess.The dynamical evolution of the systems obeys Eq. 1for g time steps, when the agents are allowed to exchangeplaces in the ring in order to help them reaching a defi-nite decision ( | x i | = x lim ). This is done according to thefollowing rules: One chooses N pairs of agents at ran-dom, and picks up the pairs of agents with both of thembeing not decided ( | x i | , | x j | < x lim ). For these pairs onecalculates a measure of the distance between the agents’opinions p ( m ) ij = 14 x lim (cid:104) | x i − (cid:104) x (cid:105) ( m ) i | + | x j − (cid:104) x (cid:105) ( m ) j | (cid:105) , (4)and compares it with the same quantity if one exchanges i and j , namely q ( m ) ij = 14 x lim (cid:104) | x i − (cid:104) x (cid:105) ( m ) j | + | x j − (cid:104) x (cid:105) ( m ) i | (cid:105) . (5)If p ( m ) ij > q ( m ) ij one exchanges places. In the aboveformulas (cid:104) ... (cid:105) ( m ) i means the average over the m left andright neighbors of site i , as in Eq. 2. This procedure isrepeated every g time steps, until one is not able to findfavorable changes or all the agents have reached eitherone of the two limit opinions, since the exchange processonly deals with pairs of undecided agents. Observe thatthese rules tend to increase opinion homogeneity in thesystem, which is reminiscent of the homophily principlementioned in the previous section. A descriptive diagramof the system and the exchange process is shown in Fig. 1. i j i - 1i + 1 j - 1 j + 1 i j i - 1i + 1 j - 1 j + 1 FIG. 1: (Color online) Diagram showing the exchange processused in the model (for m = 1). The randomly chosen unde-cided nodes i and j have q (1) ij < p (1) ij , thus they are exchanged.Observe that the grey scale representing the opinion variableis more uniform after exchange. III. NUMERICAL RESULTS
We solve the model system by numerical simulations.For that the system is initialized with values of the statevariable x i (0) in the interval [ − x lim , x lim ] drawn ran-domly from the Gaussian distribution with zero meanand unit standard deviation, cut off at ± x lim and x lim =1. Likewise the attitude parameter α i for each agent waschosen randomly from a uniform distribution in the inter-val [ − α lim , α lim ] with α lim = 1 and kept fixed through-out the whole simulations.The simulations have been carried out using a two-step process: first we solve the dynamical equations ofopinion for all undecided agents by using a simple Eu-ler numerical integration with time step dt = 10 − , andthen we perform the exchange process every g time stepsaccording to the above described rules. We keep track ofthe progress of the dynamics with two counters, namelythe number of undecided agents ( n und ) and the numberof pairs that have been exchanged ( n exch ) after everyexchange process. As the agents reach the definite opin-ions, the counter n und will decrease from its initial value N to some number close to zero. The asymptotic sta-tionary value of n und is considered as the final number ofundecided agents. Since the exchanged pairs have to beundecided, the counter n exch usually stays around or be-low n und . The exchange process is realized sequentiallyand randomly such that agents can be chosen more thanonce in the same generation. However, the probability ofsuch event decays fast with N to be very rear to have aneffect on the results.In the simulations the dynamics is let to run until theexchange of agent locations takes place very rarely. Therelaxation time for this is exceedingly large, and com-paring the results of calculations with a large number ofiterations we found that after 10 transactions, the aver-aged results over 100 realizations differ by less that 0.1%.Therefore, in all the calculations presented here we haveused these numbers. Moreover, since it turned out thatsome results depend strongly on the size of the ring for small values of N , we chose to do most of the calculationson a ring of N = 5000 and for the case m = 1, i.e. theshort range being limited to nearest neighbors. Finally,the exchange process rules can be relaxed so that anyagent (decided or undecided) can be moved, yet since theexchange of decided agents is so seldom as to have anynoticeable effect on the averaged results, we chose notto do so in order to increase the speed of the algorithm,especially for late stages of the dynamics. g < s > < n und > < f α − > (c)(b)(a) FIG. 2: (Color online) (a) Average number of undecidedagents as a function of the number of transactions per gen-eration g . The purple squares are the numerical results after10 transactions and the red circles are the agents who willnever get a decision, according to a linear analysis. The corre-sponding mean field predictions are shown as continuous lines.(b) The average fraction of undecided agents with negative α (purple squares) and its corresponding mean field prediction(continuous line). (c) Average cluster size as a function of g .Observe the plateau around g c ≈ . × . In Fig. 2(a) we show the average number of undecidedagents ( (cid:104) n und (cid:105) ) in the ring as a function of the parameter g . It is clearly seen that there is a quite sharp minimumat g c ≈ . × , a critical value of g that can be pre-dicted by using the mean field analysis, see Section IV. Inthe figure we also show by red circles the expected valueof (cid:104) n und (cid:105) for t → ∞ as obtained from a linear analy-sis, and by continuous lines the corresponding mean fieldpredictions, explained below in more detail. The criti-cal value g c signals a change of phase in the system: for g > g c all the agents get decided in the limit of infinitetime, while for g < g c a finite fraction of the network re-mains undecided for arbitrarily long times. We identifythe former phase as a state of maximum relaxation, andthe latter as a frustrated state where many agents cannotreach the limit opinions.In Fig. 2(b) it is seen that the average value of thefraction of undecided agents with negative α ( (cid:104) f − α (cid:105) ) alsoshows a sharp change of behavior, predicted by meanfield as a continuous line. For g > g c the value is 1/2,that is, the undecided agents have positive and negative α ’s indistinctly, but for sufficiently small g -values mostundecided agents have α <
0. Such phase change be-havior is also structural as evidenced in Fig. 2(c) wherethe average cluster size ( (cid:104) s (cid:105) ) remains constant around thecritical value g c . Here we have defined a cluster as a set ofconnected agents having opinions of the same sign, inde-pendent whether they are decided or not. The maximumobserved in Fig. 2(c) and likely the minimum in Fig. 2(a)are due to relaxation problems. This can be understooddue to the dynamics being stopped at a fixed time for all g , which is not enough to reach the asymptotic state. t < | x | > −101 x i −101 10 −101 t FIG. 3: (Color online) Left panel: Time evolution of the abso-lute value of opinion averaged over 100 realizations of the ringof N = 5000 agents for g → ∞ . The corresponding mean fieldprediction is shown as a continuous line. Right panels: Timehistory for a sample of 40 agents in a single realization forthree different values of g = 7 × , × , × (orderedfrom top to bottom), chosen to correspond to g < g c , g ∼ g c ,and g > g c , around g c obtained from Fig. 2. The verticaldotted lines indicate the moments when there is a change oftime regime in the dynamics, as explained in the text. On the left hand side of Fig. 3 we present the results forthe relaxation dynamics, by plotting the average absolutevalue of the state variable x as a function of time, whenthe exchange process is off ( g → ∞ ). The results areaverages over 100 realizations. It is clearly noticeablethat there are three different time regimes, seen as ans-shape curve and predicted by a mean field treatment(see Section IV). Up to around t ≈ g c (in units of dt )the evolution of (cid:104)| x |(cid:105) is very slow; between this value and τ ≈ . × the curve is concave upwards; finally forlong times the variable approaches the asymptotic value x lim = 1 very slowly and the curve is concave downwards.Only the evolution up to 10 time steps is shown, where (cid:104)| x |(cid:105) ≈ . g . If g < g c (top plot) the exchanges happen in the initial time regime, up to t ≈ g c . There-fore the evolution of the system cannot reach the relaxedstate, and frustration appears in the form of indefinitelyundecided agents. On the other hand, if g (cid:38) g c (mid-dle and bottom plots) the relaxation is already advancedwhen the exchanges are carried out, which contribute tofurther relaxation. The fact that there is a minimum inthe number of undecided nodes is due to the slow relax-ation for larger g values.The separation of the three time regimes mentionedabove is even clearer in these plots: 1) in the first timeregime practically all the agents change their opinionsslowly regardless of the value of g , so t ≈ g c can be recog-nized as the characteristic time for the first agents in thenetwork to get decided; 2) in the second time regime thedynamics speeds up exponentially and most agents getdecided (with individual trajectories getting smoother as g increases due to less frequent exchanges); 3) in thethird and final time regime only some agents remain un-decided, which for g < g c occupy frustrated regions inthe network and will be undecided indefinitely, and for g > g c will get decided after a large but finite amountof time. These remarks are supported by the analyticaltreatment in Section IV. −5 −4 −2 −1 s P ( s ) N ( < α > ) −1 −0.5 0.5 1010002000 < α > FIG. 4: (Color online) In the left hand side we show a semi-log plot of the cluster size distribution for three values of g = 7 × , × , × (green squares, purple circles andlight blue diamonds respectively). The distribution for theinitial random ring is shown for comparison as a dotted blueline. The panels on the right hand side show the number dis-tribution of α for 100 realizations of the same three values of g (ordered from top to bottom) and for three different rangesof cluster size s : red circles for s ∈ [1 , s ∈ [5 , s ∈ [15 , The cluster size distribution in the initial random ringgoes as P ( s ) = 1 / s (e.g. the probability of having s con-secutive agents with the same sign of their initial opin-ions) and it is shown as a dotted blue line in the left panelof Fig. 4. The cluster size distribution after 10 trans-actions (also shown) turns out to change very little with g , although it is quite different from the random value.The phase change behavior seen by using (cid:104) n und (cid:105) as an“order parameter” is also reflected in the preferred valueof α for clusters of different size. In the right panels ofFig. 4 we show this effect by plotting the number distri-bution of α ( N ( (cid:104) α (cid:105) )) for the three values of g , i.e. g < g c , g ∼ g c , and g > g c , and for three different ranges of clus-ter size. Observe that for g < g c small clusters (of size1 to 4) are composed mainly of agents with negative α ,clusters of medium size (5 to 14) present a bimodal distri-bution of negative and positive values, and large clusters(15 to 5000) have agents with positive α . For g aroundthe critical value the picture changes dramatically andlarge clusters start having agents with α <
0. For g > g c the number distribution of α approaches a Gaussian formindependently of the cluster size.So far we have considered the case where the shortrange interaction deals with nearest neighbors only ( m =1). We have also studied the situation in which the shortrange interaction includes the second neighbors, i.e. m =2. In this case a phase change behavior is also clearlyvisible in the number of undecided agents as a functionof g , though the position where it appears has movedslightly. IV. MEAN FIELD CALCULATIONS
In this section we shall investigate the peculiar featuresof the phase change exhibited by our model, namely, thereasons why some agents are undecided, the factors thatdetermine the average cluster size, and the peculiar dis-tribution of agents with negative attitude parameter α in the final network configuration. For this we shall per-form a linear analysis of the dynamics and introduce somemean field ideas that may help understanding the role ofthe different time scales and their effect on the structureof the network, in particular the role of parameter g . A. Linear analysis
The quantity that here plays the role of “order param-eter” is the number of undecided agents. However, inany long but finite numerical calculation, out of the to-tal number of agents that appear as undecided (see thepurple squares in Fig. 2(a)) only a fraction will remainundecided forever. We shall investigate first the circum-stances that prevent agents to reach a limit opinion.There are different scenarios depending on the valuesof f s and f l in Eq. 1. When the long range term f l = 0,Eq. 1 has a simple exponential solution and the only situ-ation that prevents the limit value sgn( x i ) to be reachedis when the short range term f s ≤
0, so the agent remainsundecided forever. If f s = 0 and f l (cid:54) = 0 the solution islinear in time and the agent will eventually reach a limitvalue. Notice that for m = 1 (i.e. short range interactionwith nearest neighbors only) this situation correspondsto an agent at the border between two groups of oppositeopinion, and once that agent becomes decided the borderis displaced by one site. Since the ring is symmetric, the net displacement of the border will be zero, and this willgive a characteristic cluster size.Eq. 1 exhibits various fixed points, on top of the limitvalues x i = ±
1. For each agent i there is a fixed point at | x ,i | <
1, where x ,i = − f l α i f s . (6)If both f s and f l (cid:54) = 0, one can perform a linear stabilityanalysis around the fixed point of Eq. 6. Then, agent i is considered indefinitely undecided if this fixed point isstable, that is, when the real part of the eigenvalue λ i = ∂ ( ∂ t x i ) ∂x i (cid:12)(cid:12)(cid:12)(cid:12) x i = x ,i = (cid:104) x (cid:105) (1) i [2 θ ( x ,i ) − , (7)is negative. In this equation θ ( x ,i ) is the Heavisidestep function. It should be noted that the occurrenceof (cid:60) [ λ i ] < g → ∞ ). The reason is that sgn( x ,i ) mustbe opposite to the sign of (cid:104) x (cid:105) (1) i , which has to be differentfrom zero and eventually ±
1. This means that the agentis embedded in a very adverse environment of immediateneighbors, a situation not favored by the dynamics thattends to diminish disagreement between the agent andits first neighbors. The only possibility for an agent toremain undecided forever is when the magnitude of α i islarge enough to hamper the dynamics. However, the α distribution is flat and the probability for this to happenis of the order of O (1 /N ).Summarizing, an agent can only be undecided in thelimit of t → ∞ if:(a) f l = 0 and f s ≤
0, or if(b) f s , f l (cid:54) = 0 and (cid:60) [ λ i ] < (cid:104) n und (cid:105) after 10 transactions (shownin Fig. 2(a) as purple squares), we have tested all agentsthat fullfil any of these two conditions to remain unde-cided forever, and plotted their numbers in the figure asred circles. Indeed, the asymptotic number of undecidedagents is nonzero for g < g c and zero for g > g c . Thelatter is in agreement with our previous mean field pre-diction, drawn as a continuous red line in Fig. 2(a).We now investigate the form of the curve for the num-ber of undecided agents for g < g c , which can be esti-mated from the initial Gaussian distribution of x . First,the symmetry of sign in the distribution of α implies thatonly half of the agents are likely to have α < x = x (0) e t/g c before the exchange process takesplace, at least those agents with initial | x (0) | < x g re-main undecided at t = g , where x g = e − g/g c . Therefore,the number of undecided agents as a function of g can becalculated from the initial distribution of x as n und ( g ) = N e − g/ ( g c √ / √ − erf( e − / √ / √ / √
2) (8)where erf( x ) = (2 / √ π ) (cid:82) x e − u du is the error function,and the factor of 1/2 is due to the sign symmetry. Theresult of Eq. 8 is plotted in Fig. 2(a) as a purple line,where the value g c ≈ . × has been fitted with least-squares technique. Notice that the agreement with thecalculation (purple squares) is considerably good. Thetheoretical estimation of the truly undecided agents (redcircles in the figure) is more involved, since the actionsof exchanging become important, and this will be thematter of further study. B. Mean field for (cid:104)| x |(cid:105) The time evolution of the average absolute value ofopinion in the network when there are no exchanges canbe understood by an estimation of the characteristic time( τ ) for the whole system to reach the limit values of opin-ion. This is done assuming that g (cid:38) τ , where one can byuse a mean field approach similar to the one described inour previous model [28]. Although the network topologythere is different, the mechanisms that result in the mag-netization relaxation of all ferromagnetic-like problemsare similar. The average number of undecided agents asa function of time is found to be (cid:104) n und ( t ) (cid:105) = N − ( N + 1) coth (cid:18) N + 12 tτ (cid:19) + coth (cid:18) t τ (cid:19) , (9)where the time scale τ is related to the critical value g c as τ = g c N/
40, see [28]. We now follow a procedure similarto that of the previous subsection, without consideringexchange processes. Since Eq. 1 has an approximate so-lution x = x (0) e t/τ , only the agents with initial opinion | x (0) | > x t = e − t/τ can get decided at time t , while therest of the agents are still undecided. Then the averageabsolute opinion of the decided agents is 2 (cid:82) x t P ( x ) dx and that of the undecided agents is 2 e t/τ (cid:82) x t xP ( x ) dx .By integrating the distribution of initial opinions P ( x )we get (cid:104)| x ( t ) |(cid:105) = 1 − erf( x t / √ / √
2) + (cid:112) /π erf(1 / √ e t/τ (cid:16) − e − x t / (cid:17) . (10)Eq. 10 has been fitted to the numerical results shownin the left panel of Fig. 3 with least-squares technique,giving a value of τ ≈ . × , which in turn correspondsto g c = 40 τ /N ≈ . × . This is in good agreementwith our estimate of last subsection and with the value inFig. 2(a) of g at the minimum in the number of undecidedagents after a finite number of transactions. Moreover,the slope of Eq. 10 is (cid:104)| x |(cid:105) (cid:48) ≈ × − for 0 < t < g c ,then it drops fast at around t ≈ τ and is asymptoticallyzero for t (cid:29) τ . This illustrates the three time regimes ofopinion evolution discussed in Section III and indicatedin Fig. 3 as vertical dotted lines. We detected that thefitting of Eq. 10 is very good for short times but starts todeviate significantly for longer times, in a similar fash-ion as the approximation for the fraction of undecided agents in [28]. This is to be expected, since in this meanfield approach we have not taken into account the effectsproduced by the random distribution of α . As a con-sequence Eq. 10 relaxes faster to the asymptotic state (cid:104)| x |(cid:105) = 1 than the actual dynamics. C. Analysis of (cid:104) f − α (cid:105) and (cid:104) s (cid:105) From the subset of undecided agents after a long but fi-nite time we can also calculate the fraction of agents withnegative α . For g > g c , the symmetry of sign in Eq. 1 andin the initial x and α distributions implies that α shouldbe distributed evenly among all agents and a value of (cid:104) f − α (cid:105) = 1 / g < g c all undecided agents have α <
0. This fact, although apparently logical, is puzzling,since it holds even for reasonably large values of g (up to10 ), but it can be explained as follows: After runningthe dynamics for a long time f l is a very small number,since the average overall opinion approaches zero, and thesecond term of Eq. 1 is no longer important. Therefore,the only way that agent i avoids the exponential approachto a limit opinion and remains undecided is that it findsitself in an adverse environment, such that it is likelyto be chosen for an exchange many times. Furthermore,the exchanges have to modify the tendency of the agenttowards a given limit opinion constantly. Remember thatthe condition to be chosen for exchange is p (1) ij > q (1) ij ,meaning that the opinion in the neighborhoods of agents i and j is more homogeneous after the exchange.We now show that only the agents with negative α canbe in this situation after a large number of exchangeshave taken place. Consider an exchange process betweentwo undecided agents with the same sign in their attitudeparameter. If α i is positive the dynamics of Eq. 1 makesit likely that agent i is surrounded by neighbors thatshare its own opinion, and since agent j is in the sameconditions as agent i , one infers from Eq. 4 that p (1) ij ≈ q (1) ij > i and j arevery seldom chosen for a location exchange, eventuallyreaching limit values of opinion. On the other hand, if α i is negative it is likely that x i and (cid:104) x (cid:105) (1) i have oppositesigns, and a similar effect in the neighborhood of agent j results in p (1) ij >
0. Since approximately half of the agents j share the same sign as the neighbors of i and viceversa, q (1) ij ≈ p (1) ij ≈ q (1) ij and the exchanges are not as common as when both α i and α j are negative. Therefore, a negative attitudeparameter along with the existence of many exchangeprocesses hampers the possibility of agents attaining adefinite decision. This result is plotted in the left part ofFig. 2(b) as a horizontal purple line.As a corollary of this analysis, we can anticipate thestructure of a typical configuration of opinions in the sys-tem after a large but finite number of transactions. For g < g c the undecided agents with α < g > g c most undecidedagents have f s = 0 and evolve slowly and linearly towardsa limit opinion, therefore they should be at the borders ofclusters with different definite opinions. This is truly thecase, as can be seen by comparing the purple squares onthe right hand side of Fig. 2(a) with the correspondingpurple squares of Fig. 2(c). Indeed, the average clustersize is (cid:104) s (cid:105) ≈
10 for large g , and the number of undecidedagents detected in the calculation is (cid:104) n und (cid:105) ≈ N/ (cid:104) s (cid:105) . V. DISCUSSION
In this paper we studied the coevolution of opinionsand the embedding of individuals in their environment.For the opinion dynamics we adopted earlier introducedcontinuous state variable equations [28], that includeshort range ferromagnetic interactions for describing ho-mophily between neighboring agents, and long range in-teractions for describing how the overall mood of the ma-jority affects the agent modulated by its attitude parame-ter being either positive or negative. This opinion updategives rise to short time scale transaction dynamics. Forthe model geometry or connectivity between agents, weused ring topology instead of a more complex networktopology, we studied earlier [28, 29]. The long or slowtime scale dynamics of environment changes was carriedout by exchanging the locations of pairs of agents. Thesetwo time scales are then well separated and their relationserves as a control parameter.As the main result of our study we find that by varyingthe time-scale parameter there is a phase change in thenumber of undecided individuals, which turned out to bemainly driven by the environment exchange dynamics.In order to understand this effect the following shouldbe noted. First there is competing interaction due tothe negative α ’s. Second, due to the asymmetry betweenthe long and short range interactions (since only the lat-ter are considered for exchanges), this competition doesnot lead to permanent frustration, provided that enoughtime is given for relaxation. However, if the relaxation ishampered by too frequent changes in the neighborhoodas well as by not allowing enough exchanges to find theglobal optimum, frustration appears as a nonzero numberof indefinitely undecided agents. Thus the phase changebehavior is due to the separation of time scales and dueto insufficient relaxation. The mean field analysis whichwe performed for the system supports the above picture. It should be noted that there are relevant similaritiesand differences between the ring model studied here andour previous network model [28]. First, the transactiondynamics defined by Eq. 1 is equivalent in both mod-els, therefore producing similar relaxation processes inthe limit of large g that can be studied with the samemean field approach, as has been discussed in Section IV.Second, the exchange process used in the ring model issystematically different from the rewiring framework ofthe network model, since it changes the opinion distri-bution in the system by keeping the topology constant,thus making both models fundamentally distinct. Fur-thermore, the exchange process used here has some soci-ological background with the concept of homophily, sinceit favors homogeneity of opinion in the ring, but it is sim-ple enough as to allow a deeper mean field treatment thanthe one performed for the network model.Even though the exchange process differs from therewiring scheme treated in [29], both models can producesmall clusters of agents with α < α > g , as shownin Fig. 4. The mean field performed in the simpler ringtopology suggests that a low time-scale ratio is responsi-ble for frustration that creates small groups of undecidedagents with negative α , while a full relaxation tends todestroy this structure. In the case of the network modelthe rewiring rules would then be the sources of frustra-tion, and an appropriate value of g could freeze agentswith α < Acknowledgments
G.I. and K.K. acknowledge the Academy of Finland,the Finnish Center of Excellence program 2006 - 2011,under Project No. 129670. K.K. and J.K. acknowl-edge support from EU’s FP7 FET Open STREP ProjectICTeCollective No. 238597 and J.K. also support byFinland Distinguished Professor (FiDiPro) program ofTEKES. K.K. and R.A.B. want to acknowledge finan-cial support from Conacyt through Project No. 79641. R.A.B. is grateful to the Centre of Excellence in Com-putational Complex Systems Research - COSY of AaltoUniversity for support and hospitality for the visits whenmost of this work has been done. [1] R. A. Baron, N. R. Branscombe, and D. R. Byrne,
SocialPsychology (Pearson International, Boston, MA, 2010).[2] F. Wu and B. A. Huberman (2004), e-print arXiv:cond-mat/0407252v3.[3] D. J. Watts and P. S. Dodds, J. Cons. Res. , 441(2007).[4] H. White, S. Boorman, and R. Breiger, Am. J. Sociol. , 730 (1976).[5] C. Castellano, S. Fortunato, and V. Loreto, Rev. Mod.Phys. , 591 (2009).[6] P. Sobkowicz, J. Artif. Soc. Soc. Simul. , 11 (2009).[7] L. P. Kadanoff, Statistical Physics: Statics, Dynamicsand Renormalization (World Scientific, Singapore, 2000).[8] J. A. Ho(cid:32)lyst, K. Kacperski, and F. Schweitzer, Annu.Rev. Comput. Phys. , 253 (2001).[9] R. Hegselmann and U. Krause, J. Artif. Soc. Soc. Simul. , 2 (2002).[10] R. A. Holley and T. M. Liggett, Ann. Probab. , 643(1975).[11] W. Weidlich, Phys. Rep. , 1 (1991).[12] K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C , 1157 (2000).[13] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch,Adv. Complex Syst. , 87 (2000).[14] F. Vazquez, P. L. Krapivsky, and S. Redner, J. Phys. A , L61 (2003).[15] T. Gross and B. Blasius, J. R. Soc. Interface , 259(2008).[16] M. Perc and A. Szolnoki, BioSystems , 109 (2010).[17] M. G. Zimmermann, V. M. Egu´ıluz, and M. San Miguel,Phys. Rev. E , 065102(R) (2004).[18] G. C. M. A. Ehrhardt, M. Marsili, and F. Vega-Redondo,Phys. Rev. E , 036106 (2006).[19] F. Vazquez, J. C. Gonz´alez-Avella, V. M. Egu´ıluz, andM. San Miguel, Phys. Rev. E , 046120 (2007). [20] J. M. Pacheco, A. Traulsen, H. Ohtsuki, and M. A.Nowak, J. Theor. Biol. , 723 (2008).[21] J. Poncela, J. G´omez-Garde˜nes, L. M. Flor´ıa, A. S´anchez,and Y. Moreno, PLoS ONE , 2449 (2008).[22] S. Gil and D. H. Zanette, Phys. Lett. A , 89 (2006).[23] K. Suchecki, V. M. Egu´ıluz, and M. San Miguel, Phys.Rev. E , 036132 (2005).[24] P. Holme and M. E. J. Newman, Phys. Rev. E , 056108(2006).[25] I. J. Benczik, S. Z. Benczik, B. Schmittmann, andR. K. P. Zia, Europhys. Lett. , 48006 (2008).[26] C. Nardini, B. Kozma, and A. Barrat, Phys. Rev. Lett. , 158701 (2008).[27] F. Vazquez, V. M. Egu´ıluz, and M. San Miguel, Phys.Rev. Lett. , 108702 (2008).[28] G. I˜niguez, J. Kert´esz, K. K. Kaski, and R. A. Barrio,Phys. Rev. E , 066119 (2009).[29] G. I˜niguez, R. A. Barrio, J. Kert´esz, and K. K. Kaski(2010), doi:10.1016/j.cpc.2010.11.020 (In Press).[30] M. Mezard, G. Parisi, and M. Virasoro, Spin Glass The-ory and Beyond (Word Scientific, Singapore, 1987).[31] T. C. Schelling, J. Math. Sociol. , 143 (1971).[32] T. C. Schelling, Micromotives and Macrobehavior (W.W. Norton, New York, 1978).[33] J. Zhang, J. of Economic Behavior & Org. , 533 (2004).[34] D. Vinkovi´c and A. Kirman, Proc. Natl. Acad. Sci.U.S.A. , 19261 (2006).[35] D. Stauffer and S. Solomon, Eur. Phys. J. B , 473(2007).[36] L. Dall’Asta, C. Castellano, and M. Marsili, J. Stat.Mech. , L07002 (2008).[37] L. Gauvin, J. Vannimenus, and J.-P. Nadal, Eur. Phys.J. B70