Eur. Phys.J. B 65, 251–255 (2008) THE EUROPEAN DOI:10.1140/epjb/e2008-00342-3 PHYSICAL JOURNAL B Opinion dynamics on directed small-world networks L.-L. Jiang1,a, D.-Y. Hua2, J.-F. Zhu1, B.-H. Wang1,b, and T. Zhou1,3 1 Department of Modern Physics, Universityof Science and Technology of China, Hefei 230026, P.R.China 2 Department of Physics, NingboUniversity,Ningbo Zhejiang 315211, P.R.China 3 Department of Physics, Universityof Fribourg, Chemin du Muse 3, 1700 Fribourg, Switzerland Received 18 January 2008 / Received in final form 22 July 2008 Published online 10 September2008 – (cid:2)c EDP Sciences, Societ`a Italiana diFisica, Springer-Verlag2008 Abstract. Inthispaper,weinvestigatetheself-affirmationeffectonformationofpublicopinioninadirected small-world social network. The system presents a non-equilibrium phase transition from a consensus state to a disordered state with coexistence of opinions. The dynamical behaviors are very sensitive to the density of long-range-directed interactions and the strength of self-affirmation. When the long-range- directedinteractionsaresparseandindividualgenerallydoesnotinsistonhis/heropinion,thesystemwill display a continuous phase transition, in the opposite case with strong self-affirmation and dense long- range-directed interactions, the system does not display a phase transition. Between those two extreme cases, the system undergoes a discontinuousphase transition. PACS. 89.75.-k Complex systems – 89.65.-s Social and economic systems – 05.70.Fh Phase transitions: general studies – 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) 1 Introduction to account for the self-affirmation effect of individuals as well as directed relations between agents. Recently, much effort has been devoted to the studying In the realworld,interactions between individuals are opiniondynamics[1–3].Statisticalphysicsprovidesquan- notonlyshortranged,butalsolongranged[21,22].Thein- titative tools to reveal the underlying laws that govern teractionusuallydisplaysadirectedfeature,whichmeans the opinion dynamics [4]. Agent-based models have been an individual who receives influence from a provider may proposed to study complex phenomena of opinion forma- notaffecttheprovider.Weapplythedirectedsmall-world tion. With process of social influence [5,6], consensus in networks proposed by S´anchez et al. [14] to represent opinion formation achieves. For example, the opinion of this kind of relations between individuals. In this paper, anindividual may be affected by its nearestneighbors,as we present an opinion dynamics model including individ- described in the Sznajd model [7,8], the Galam’s major- ual self-affirmation psychological feature and long-range- ity rule [9,10], and the Axelrod multicultural model [11]. directed correlations between individuals. The main dif- On the other hand, the contrarian effect is introduced to ference between this model and other physics’ inspired account for the phenomenon of a transition from a polar- models is that this model takes into account both effects ized opinion state to a coexistent opinions state [12,13]. of self-affirmation and social structure. The former lies in The similar results are also obtained in references [14–16] themicroscopiclevel,whilethelatterconcernsthemacro- in which social temperature is considered. The real-life scopic impacts. The parameter space can be roughly di- systemoftenseems ablackboxtous:the outcomecanbe videdintothreeregions,inwhich,respectively,weobserve observed, but the hidden mechanism is not visible. If we continuous phase transition, discontinuous phase transi- see many individuals hold the same opinion, we say The tion and no phase transition. Spiral of Silence phenomenon[17]occurs.Itiscommonin real world that people adhere to their own opinion even opposite to most of their friends [18–20], which we call 2 Model self-affirmation of individuals, similar to the contrarian effect [12,13]. Inour early work,the influence of inflexible In this section, we introduce a directed small-world net- units has been investigated in a simple social model [16]. work model and an opinion dynamics model. We start It is found that this kind of effect canlead to a nontrivial with a two-dimensional regular lattice, in which every phase diagram. However, traditional opinion models fail node is connected with adjacent four nodes inwardly and a e-mail: [email protected] outwardly respectively, then, with probability p, rewire b e-mail: [email protected] eachoutwardlinktoarandomlychosennonadjacentnode. 252 The European Physical Journal B 3 Simulations In order to describe the evolution process of the model, we employ a magnetization-like order parameter (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:5)L2 (cid:4) m=(cid:4)(cid:4)(cid:4)L12 σi(cid:4)(cid:4)(cid:4),σi ∈{+1,−1}. (2) i=1 The network size is L × L and m is the absolute aver- age value of the states of all nodes. An extensive Monte Carlo numerical simulation has been performed on our model with a random initial configuration and a periodic boundary.Results are calculated after the systemreaches a non-equilibriumstationarystate. Inorder to reduce the occasionalerrors,fornetworksizeL=16,32,64,and100, Fig. 1. Illustration of the structure of a directed small-world wehaveaveragedthe resultover40000,10000,2000,and network for p=0.1 [14]. 1000 runs, respectively, with different network structures under different random initial configurations. Obviously, when (cid:4)m(cid:5) tends to 1, the system enters into an ordered Inthisway,asshowninFigure1,adirectednetworkwith state, i.e., individuals in the system reach a consensus adensitypoflong-range-directedlinksisobtained.Inthis opinion. Meanwhile, if the system stays in a disordered network, nodes represent individuals in the social system state, the order parameter scales as (cid:4)m(cid:5) ∼ 1. As shown and the outward links represent the influences from oth- L in Figure 2, the system reaches an ordered state when T ers.Eachnode connects with four nodes outwardlywhich are called its mates. islessthanacriticaltemperatureTc.Thesystemdisplays a continuous phase transition for p = 0.1 and q = 0.9, Inthenetwork,anindividualstaterepresentsitsview- while a discontinuous phase transition for p = 0.9 and point, which evolves according to the social process, de- q = 0.9. From the probability density functions (PDFs) terminednotonlybyothercorrelativesurroundingeffects of the order parameter near the phase transition point but also by its own character. It is supposed that there of the phase diagram (p,T), one can distinguish between are two kinds of possible opinions in the system, just as the continuous phase transition and discontinuous phase the agreementand disagreementin the election,and each transition clearly. According to PDFs inserted in the up- individual takes only one of them. Therefore, the state of a node i can be described as σi, σi ∈ {+1,−1}. We de- perandlowerpanelsofFigure2,itisfoundthatthemost probable values of m, which correspond to the highest scrib(cid:2)e the difference of σi from its mates by W(σi) = 4 peaks of PDFs, jump little from nonzero to zero in the 2σi j=1σj, where σj (j = 1,2,3,4) are states of i’s continuous phase transition from Figure 2a to Figure 2b, mates. In addition, q (0 < q ≤ 1) is used to describe while sharply in the discontinuous one from Figure 2c to theprobability,withwhichindividualsfollowtheirmates’ Figure 2d. It seems that the long-range correlations can dominant opinion. Meanwhile 1 − q represents the self- change the nature of phase transition. affirmationprobabilityofindividuals,with whichanindi- Evidently, given q = 0.9, the system varies from the vidualinsistsonhis/herownopinionthoughitisopposite continuousphasetransitiontodiscontinuousphasetransi- to the majority of his/her mates. tion when the density of long-range-directed connections According to the illuminationabove,we introduce the is high enough. It is natural to ask how these topology dynamical rule as follows: W(σi) > 0 indicates that σi structures influence the opinion dynamics. To solve this is the same as the majority of σj (j = 1,2,3,4) and σi problem, we define the domain size s as the number of overturnswithprobabilityexp[−W(σi)/T]whichdepends neighborhood nodes in the same state. As shown in Fig- on a temperature-like parameter T. W(σi) < 0 indicates ure 3a, it is found that the domain size s distributes in a that σi is opposite to the majority of σj (j = 1,2,3,4), power law, g(s) ∼ s−τ for s (cid:7) L2 at the critical point, andσi overturnswithprobabilityq.WhenW(σi)=0,the where g(s) is the probability function. One can find that state of node i overturns with probability q also. So that there is a localmaximumprobability of largedomainsize the overturning probability P(σi) of σi is given by forp=0.1.Smallerpindicatesmorelocalizedinteractions (cid:3) exp[−W(σi)/T], for W(σi)>0 between individuals, and a large domain emerges more easily. Besides, we calculate the number of time steps, t, P(σi)= (1) q, for W(σi)≤0. during which an individual holds the same opinion. As shown in Figure 3b, one can find that individuals change Fromthedynamicalrule(1),wecanseethat,whenq =1, their own opinion for p = 0.9 more frequently than for the current model restores to the network-based Ising p = 0.1 at T = 0.1, and the probability of t obeys a model [14]. However, our model is non-equilibrium be- power-law distribution f(t) ∼ t−γ (t < t0) for p = 0.1. cause the overturning probability of a state does not sat- Clearly, p plays the key role in determining the commu- isfy the detailed equilibrium condition. nication strength between different opinion domains, and L.-L. Jiang et al.: Opinion dynamics on directed small-world networks 253 Fig.2. (cid:3)m(cid:4)varieswithT fordifferentsystemsizes.Theupper andlowerplotsareforp=0.1andp=0.9,withq=0.9fixed. Insets are PDFs nearby the phase transition point: (a) T → Tc−,(b) T →Tc+,(c) T →Tc−, (d) T →Tc+. Fig. 3. (Color online) Distributions of domain size g(s) (a) and opinion holding time f(t) (b) in different networks with individualschangetheirownopinionsmorefrequentlydue q = 0.9 fixed, (a) for T = Tc and (b) for T = 0.1. The data to the long-range connections between different opinion points are obtained from 105 samples with fixed network size, domains. L=64. Figures 4a–4c show the phase diagram of opinion dy- namics determined by the network structure parameter p exponent in the space direction. At critical point, various as well as the individual self-affirmation psychology char- acteristic parameter 1−q. In Figure 4a for q = 0.9, the ensemble-averaged quantities depend on the ratio of sys- tem size and the correlation length L/ξ. Therefore, the swyhsitleemdidscisopnltaiynsuocuosntpihnuasoeustrpanhsaisteiotnrafonrsiptio≥npfco.rTphe<sypsc-, orderparameter(cid:4)m(cid:5)satisfiesthescalinglawintheneigh- tem displays discontinuous phase transitionfor q =0.5 in borhoodofthecriticalpoint:(cid:4)m(cid:5)∝L−β/νf[(Tc−T)L1/ν]. Figure 4b. The system displays discontinuous phase tran- At Tc, (cid:4)m(cid:5) ∝ L−β/ν, and we obtain β/ν = 0.530(5) for sition for p < p0 and q = 0.3 in Figure 4c, while the p = 0.1 and q = 0.9 in Figure 5a. Figure 5b reports system does not have a phase transition for p > p0 and (cid:4)m(cid:5)Lβ/ν versus (1−T/Tc)L1/ν on a double-logarithmic q =0.3 in Figure 4c. As shown in Figure 4d, the continu- plot for q = 0.1 and q = 0.9. It is shown that with the ous phase transitiontakes place in the areaI, the discon- choices β/ν = 0.530(5) and ν = 0.92(1) the data for dif- tinuous phase transition appears in the area II and the ferent network sizes are well collapsed on a single master system stays in disordered state without phase transition curve[23].Theslopeofthelineisβ =0.488±0.005,which in the area III. When both the parameters p and 1−q gives the asymptotic behavior for (cid:4)m(cid:5)Lβ/ν as L→∞. So arelargeenough,indicatingweakinteractionsbetweenin- that, we have β = 0.488(5), ν = 0.92(1) for p = 0.1 and dividualsinbothlocalandgloballevels,the systemkeeps q =0.9. Comparing to critical exponents of 0.50 and 0.94 disordered at any temperature, i.e., the phase transition for p=0.1 and q =1.0 in reference [14], 0.30and 0.80for can not take place in the system, as in the area III of p = 0.5 and q = 1.0 also in reference [14], 0.118 and 0.8 Figure 4d. for p=0.0 and q =0.8 in reference [16], 0.11and 0.85for A finite-size scaling analysis is employed to study the p=0.0andq =0.6alsoinreference[16],wecanconclude criticalbehaviorofcontinuousphasetransitionforp=0.1 that critical exponents β and ν depend on both p and q. and q = 0.9. In the neighborhood of the critical point Tc, (cid:4)m(cid:5) ∝ (Tc − T)β, (T < Tc), where β is the order 4 Conclusion parameter exponent. Besides, when T is near to critical point Tc of the second order phase transition, a charac- ter length scale ξ denotes the correlation length in space. In conclusion, the effect of long-range-directed links ξ ∝(Tc−T)−ν, (T <Tc), where ν is a correlationlength between individuals on the opinion formation is 254 The European Physical Journal B Fig. 4. Thephasediagram ofthemodelinthep−q plane.PointsarenumericaldeterminationsofthecriticaltemperaturesTc fordifferentp.Theopencirclescorrespondtocontinuoustransition,whilethesolidonescorrespondtodiscontinuoustransition. Plot (d)reportsthephasediagram: thesystemdisplayscontinuousphasetransition inregion I,discontinuousphasetransition in region II,and no phase transition in region III. systematically explored. The results show that the sys- tem takes on a non-equilibrium phase transition from a consensus state to a state of coexistence of different opin- ions.With increasingdensity oflong-range-directedlinks, acontinuousphasetransitionchangesintoadiscontinuous one. The reason why the phase transition behavior varies is that the long-rangelinks make individuals change their ownopinionsmorefrequently.Itisworthmentioningthat thesystemkeepsinadisorderedstatewhentherearesuffi- cientlong-rangelinks.Thoselong-rangeinteractionsbreak the possiblylocalorder,thus hinderthe globalconsensus. Thesimilarphenomenonoforder-disordernonequilibrium phasetransitionemerginginthesystemisalsoobservedin contrarians’ models [12,13,15] or a non-conservative vot- ers’ model [24]. Phase transitions from a consensus state to a dis- ν β ordered state are common features of opinion dynam- ics seized by both contrarians’ models and the present model. A contrarian is defined as an agent adopting the choice opposite to the prevailing choice of others what- ever this choice is (see Ref. [12]), while self-affirmation effect is presented as a probability at which an agent in- sists on his/her opinion opposite to majority of his/her neighborhood. However, directionaly is not considered in ν classical contrarians’ models, though the social interac- tions between agents are, in general, not symmetric. In Fig. 5. Finite size scaling of continuous phase transition for fact,thepresentdirectedsmall-worldtopologyplaysacru- p=0.1 and q=0.9. (a) A log-log plot of theorder parameter cial rule in opinion dynamics. The present model uncov- (cid:3)m(cid:4)against L.(b)Doublelogarithmicplotof(cid:3)m(cid:4)Lβ/ν versus ers that behaviors of phase transition are simultaneously (1−T/Tc)L1/υ for L=16, 32, 64, and 100. determined by strength of self-affirmation and density of L.-L. Jiang et al.: Opinion dynamics on directed small-world networks 255 long-range-directed links. In addition, a new kind of fan- 8. D. Stauffer, P.M.C. de Oliveira, Eur. Phys. J. B 30, 587 tastic phenomena is observed in the present model: re- (2002) sulting from the coupling effects of strong self-affirmation 9. S. 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