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Interplay between media and social influence in the collective behavior of opinion dynamics

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Messages conveyed by media act as a major drive in shaping attitudes and inducing opinion shift. On the other hand, individuals are strongly affected by peer pressure while forming their own judgment. We solve a general model of opinion dynamics
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  Interplay between media and social influencein the collective behavior of opinion dynamics Francesca Colaiori 1,2 and Claudio Castellano 3,2 1 Istituto dei Sistemi Complessi (ISC-CNR), UOS Sapienza, c/o Dipartimento di Fisica,Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy  2 Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy  3 Istituto dei Sistemi Complessi (ISC-CNR), Via dei Taurini 19, 00185 Roma, Italy  (Dated: October 14, 2015)Messages conveyed by media act as a major drive in shaping attitudes and inducing opinion shift.On the other hand, individuals are strongly affected by peers’ pressure while forming their own judgment. We solve a general model of opinion dynamics where individuals either hold one out of twoalternative opinions or are undecided, and interact pairwise while exposed to an external influence.As media pressure increases, the system moves from pluralism to global consensus; four distinctclasses of collective behavior emerge, crucially depending on the outcome of direct interactions amongindividuals holding opposite opinions. Observed nontrivial behaviors include hysteretic phenomenaand resilience of minority opinions. Notably, consensus could be unachievable even when media andmicroscopic interactions are biased in favour of the same opinion: the unfavoured opinion mighteven gain the support of the majority. PACS numbers: 87.23.Ge, 89.65.Ef, 02.50.Le, 05.45.-a I. INTRODUCTION Public opinion formation is a complex mechanism:people constantly process a huge amount of informationto make their own judgment. Individuals form, recon-sider and possibly change their opinions under a steadyflow of news and a constant interchange with others. So-cial interactions have a strong effect on opinion forma-tion [1]: the pressure to conform to the opinion of othersmay in some cases even overcome empirical evidence [2]or undermine the wisdom of the crowd [3]. On the otherhand media also play a central role: they can influencepublic knowledge, attitudes and behavior by choosing theslant of a particular news story, or just by selecting whatto report. An intense activity has recently investigatedmacroscopic effects of these fundamental mechanisms ex-ploiting the connection between opinion dynamics andsimple non–equilibrium statistical physics models [4, 5].Media influence has been considered in the literature,in particular within the framework of cultural dynam-ics [6–8] and continuous opinion dynamics with hetero-geneous confidence bounds [9–12]. The fundamental caseof binary opinions has received less attention for whatconcerns the role of media [13–15], although the similarcase of proselytism by committed agents (zealots) has at-tracted considerable interest [16, 17]. The case of binaryopinions is clearly relevant to yes/no questions, but notlimited to that: when people are prompted with impor-tant questions that admit many possible answers theirattitudes tend to be polarized, most people sharing oneout of two opposite opinions [18].In this paper we address the problem of how peopleform their opinions based on the message they receive,both from traditional media and their social network,by studying a general class of opinion dynamics mod-els. At any given time, each individual in the popu-lation is either supporting one of two alternative opin-ions or “undecided”. The possibility to be in a thirdstate may have crucial effects in the case of a binarychoice [19–22]. Individuals are exposed to some externalsource of information biased towards one opinion (main-stream) and exchange information upon pairwise interac-tion: both factors may cause them to change their state.The undecided state accounts for individuals being unin-terested, uninformed, or generally confused on the givenissue. Carrying no opinion, they are assumed to havea passive role in the interactions. We account for theeffect of media in the simplest possible way by assum-ing that people have some general tendency to conformto the media recommendation. On the other hand, weconsider totally general rules for the interactions amongpairs of agents. Our general model encompasses there-fore a large class of specific models, each one identified bya given set of parameters defining the interaction rules.We study in mean-field this general model, determiningthe stationary solutions and their stability as a functionof the external bias and of the parameters specifying pairinteractions. We uncover the emergence of four distinctclasses of collective behavior, characterized by differentresponses to the media exposure. Only two linear combi-nations of the parameters defining the general dynamicsare relevant in determining which class a specific modelbelongs to. The results of numerical simulations, per-formed in systems with interaction patterns described bycomplex networks, support the general validity of the MFpicture. II. GENERAL MODEL We now define in detail the general model. Each agentcan be in one of three states: holding opinion  A , holding  2 A,A B,B U,U A,U B,U A,BA − A  1 0 0 0 0 0 B − B  0 1 0 0 0 0 U   − U   0 0 1 0 0 0 A − U ϕ 2  0 0 1 − ϕ 2  0 0 B − U   0  γ  2  0 0 1 − γ  2  0 A − B α 1  α 2  α 3  α 4  α 5  α 6 TABLE I. Each row in the table corresponds to an interaction,and each column to a possible outcome. Elements in thetable indicate the probabilities of each possible outcome forthe given interaction. In the last row  6 i =1 α i  = 1. an opposing opinion  B , or being undecided ( U  ). Agentstend to conform to the media recommendation by adopt-ing opinion  A  at a constant rate  r , independently of theircurrent state, and interact pairwise at rate  f  . In the fol-lowing we set  f   = 1 to fix the time scale. Individualsin the same state are unaltered by their mutual inter-action. In  A − U  ( B − U  ) interactions undecided agentsmay adopt, with given probability, their partner’s opin-ion, that they cannot alter: an agent holding opinion  A ( B ) has a probability  ϕ 2  ( γ  2 ) to convince the  U   agent.We assume in general  ϕ 2   =  γ  2 , allowing for  A  and  B to have unequal efficacy in persuading others. Interac-tions among agents holding opposite opinions ( A − B )may have any outcome: each of the two agents may ei-ther keep her opinion, change it to match her partner’sopinion, or get confused and turn to the undecided state,in any combination. We indicate with  { α i } 6 i =1  the prob-abilities of the six possible outcomes. Seven independentparameters fix the probabilities of each possible outcomefor any interacting pair, as summarized in Table I. Whilewe consider for simplicity symmetric roles for the two in-teracting partners, our mean–field (MF) analysis holdsmore generally, also including models that assign dis-tinct roles (e.g. speaker/listener) to the two partners.This more general case is discussed in Appendix A. Notethat asymmetric models are always equivalent, in meanfield, to their symmetrized version, the outcome of a sym-metric interaction being the average result of asymmetricinteractions with exchanged roles. III. MEAN FIELD EQUATIONS The evolution of the system is described in MF by thedynamical equations for the density of agents in eachstate:  ˙ n A  = r (1 − n A ) + 2 ϕ 1 n A n B  + 2 ϕ 2 n A (1 − n A − n B )˙ n B  = − rn B  + 2 γ  1 n A n B  + 2 γ  2 n B (1 − n A − n B ) (1)where  n A  ( n B ) denote the density of agents in the  A ( B ) state, and  ϕ 1  =  α 1  − α 2  − α 3  − α 5 ,  γ  1  =  − α 1  + α 2  − α 3  − α 4 . The density  n U   of undecided agents isobtained by normalization:  n U   = 1  −  n A  −  n B . Thedensities must belong to the  physical region   of the plane (a) ϕ  1 γ  1 0-1 10-11 FCM VCM  ZCM TC  (b) 0-1 1 γ  ∗ - ϕ  1 γ  1 r  * r  c (b) 0-1 1 γ  ∗ - ϕ  1 γ  1   r  * r  c FIG. 1. (color online)(a) Phase diagram in the ( ϕ 1 ,  γ  1 ) plane.The curve  γ  ∗ 1  separating regions  VCM   and  ZCM   depends on ϕ 2  and  γ  2  (in the figure  ϕ 2  = 0 . 1 and  γ  2  = 0 . 5). (b) Plot of  r c  (saddle–node bifurcation curve) and  r ∗ (transcritical bifur-cation line) as functions of   γ  1 , and for  ϕ 1  = − 0 . 8. ( n A ,n B ), defined by the three constraints  n A  ≥ 0,  n B  ≥ 0,  n A  +  n B  ≤  1. The coefficients  α i  appear in Eq. (1)only in two linear combinations  ϕ 1  and  γ  1  that representthe total average variation in  A  ( B ) states due to an A − B  interaction. This reduces the number of effectiveparameters from seven to four plus the external bias  r ,acting as a control parameter. The parameters  ϕ 1  and  γ  1 are bounded by  − 1 ≤ γ  1  ≤ 1,  − 1 ≤ ϕ 1  ≤ 1,  γ  1  + ϕ 1  ≤ 0(the sum  γ  1  +  ϕ 1  is minus the average net productionof undecided in an  A  −  B  interaction, that has to benon negative). Stationary solutions of Eq. (1) are theintersections of the two conic sections:  n A  [( ϕ 2 − ϕ 1 ) n B  + ϕ 2 n A − ϕ 2  + r/ 2] =  r/ 2  C  1 n B  [( γ  1 − γ  2 ) n A − γ  2 n B  + γ  2 − r/ 2] = 0  C  2 (2)The curve  C  1  is an hyperbola. One of its asymptotesis the axis  n A  = 0, so that only the upper branch  C  +1 ( n A  >  0) is physically relevant. The curve  C  2  factorizesinto the product of two lines,  R 1 , and  R 2 . The line  R 1 ( n B  = 0) always intercepts  C  +1  in  P  1  ≡ [ n B  = 0 ,n A  = 1],corresponding to the absorbing state of total consensuson opinion  A . Depending on the parameters, R 2  may in-tersect C  +1  in one, two (possibly coincident) points, eitherinside or outside the physical region, or none. Varyingthe control parameter  r , different fixed points arise andmove entering and exiting the physical region. Their flowand stability determine the collective response to the ex-ternal bias. By studying them we find the phase-diagramrepresented in Fig. 1(a), which constitutes the main re-sult of our paper. As a function of the external bias  r there are four distinct classes of collective behavior, asso-ciated with different regions of the parameter space. No-tably, the emergent behavior is essentially ruled by onlytwo of the parameters,  ϕ 1  and  γ  1 . In the next sectionwe discuss the qualitative features of each class, referringto Appendix B for analytical details on the derivation of the phase diagram. Fig. 2 represents the shapes of thecurves in Eq. (2), allowing to understand the existence  3 P 1 Q 1 P 2 Q 2 n  A n  B r=0ClassClassClassClass I  II  III  IV P 1 Q 2 Q 1 n  A n  B 0<r<r  c P 1 Q 1 =Q 2 n  A n  B r=r  c P 1 n  A n  B r>r  c P 1 Q 1 P 2 n  A n  B r=0P 1 Q 1 n  A n  B 0<r<r  * P 1 =Q 2 Q 1 n  A n  B r=r  * P 1 Q 2 Q 1 n  A n  B r  * <r<r  c P 1 Q 1 =Q 2 n  A n  B r=r  c P 1 n  A n  B r>r  c P 1 Q 1 P 2 n  A n  B r=0P 1 Q 1 n  A n  B 0<r<r  * P 1 =Q 1 n  A n  B r=r  * P 1 n  A n  B r>r  * P 1 Q 1 P 2 n  A n  B r=0P 1 n  A n  B r>0 FIG. 2. Plots of isoclines and fixed points for each class of models and each range of   r . Red solid lines are  R 1  and  R 2 , theblu dashed curve is the hyperbola  C  1 . Red stars denote stable fixed points, black circles denote saddle points, blue trianglesdenote repulsive fixed points. The shaded triangle represents the physical region  n A  ≥  0,  n B  ≥  0,  n A  + n B  ≤  1. Plots in thefirst row correspond to  ϕ 1  =  − 0 . 6,  ϕ 2  = 0 . 1,  γ  1  =  − 0 . 3,  γ  2  = 0 . 5, and  r  = 0,  r  =  r c / 2,  r  =  r c ,  r  =  r c  + 0 . 05. Plots in thesecond row correspond to  ϕ 1  = − 0 . 6,  ϕ 2  = 0 . 05,  γ  1  = 0 . 1,  γ  2  = 0 . 3, and  r  = 0,  r  =  r ∗ / 2 =  γ  1 ,  r  =  r ∗ =  γ  1 ,  r  = 0 . 2 r ∗ + 0 . 8 r c , r  =  r c ,  r  =  r c  + 0 . 05. Plots in the third row correspond to  ϕ 1  = − 0 . 5,  ϕ 2  = 0 . 7,  γ  1  = 0 . 3,  γ  2  = 0 . 5, and  r  = 0,  r  =  r ∗ / 2 =  γ  1 , r  =  r ∗ =  γ  1 ,  r  =  r ∗ + 0 . 1. Plots in the fourth row correspond to  ϕ 1  = 0 . 1,  ϕ 2  = 0 . 7,  γ  1  = − 0 . 2,  γ  2  = 0 . 5, and  r  = 0,  r  = 0 . 2. (a)P 1 n  A n  B (b)P 1 n  A n  B P 1 n  A n  B (c) FIG. 3. (color online) Phase portrait relative to Eq. (1): flows in the vector field indicate the time evolution of the system.Fixed points are the intersections between either of the two lines  R 1 ,  R 2  (red solid lines), and the curve  C  1  (blue dashed line).Black circles are saddle points, red stars are attractive fixed points. (a)  FCM   model in the region  r < r c  ( ϕ 1  = − 0 . 6,  ϕ 2  = 0 . 1, γ  1  =  − 0 . 3,  γ  2  = 0 . 5,  r  = 0 . 1); (b)  VCM   model in the range  r < r ∗ ( ϕ 1  =  − 0 . 6 ,ϕ 2  = 0 . 4,  γ  1  = 0 . 1,  γ  2  = 0 . 3,  r  = 0 . 13); (c) TC   model ( ϕ 1  = 0 . 1,  ϕ 2  = 0 . 7,  γ  1  =  − 0 . 2,  γ  2  = 0 . 5,  r  = 0 . 2). The flow is qualitatively similar to (a) for  VCM   models with r ∗ < r < r c , to (b) for  ZCM   models with  r < r c , to (c) for any class of models above  r c . See Appendix B for further details.  4  0 0.2 0.4 0.6 0.8 1 Class I: FCM  0 r  c 0.2 r n  A n  B n U  (a)Class II: VCM  0 r r  c 0.5 r  * n  A n  B n U  (b)Class III: ZCM  0 r  1 r  * n  A n  B n U  (c)Class IV: TC  01 r n  A n  B n U  (d) FIG. 4. (color online) Theoretical results for the densities of agents for realizations of each of the four classes of models: (a) Class I   model (FCM) ( ϕ 1  =  − 0 . 5,  ϕ 2  = 0 . 1,  γ  1  =  − 0 . 4, and  γ  2  = 0 . 5), (b) Class II   model (VCM) ( ϕ 1  =  − 0 . 5,  ϕ 2  = 0 . 1, γ  1  = 0 . 1, and  γ  2  = 0 . 5), (c) Class III   model (ZCM) ( ϕ 1  =  − 0 . 5,  ϕ 2  = 0 . 1,  γ  1  = 0 . 3, and  γ  2  = 0 . 5), (d) Class IV   model (TC)( ϕ 1  = 0 . 3,  ϕ 2  = 0 . 1,  γ  1  = − 0 . 5, and  γ  2  = 0 . 5). Solid (dashed) lines represent stable (unstable) lines. and positions of stationary solutions. Fig. 3 provides in-formation on flows and the stability of solutions. Fig. 4depicts the resulting behavior of the densities of agentsin the different states as a function of   r . IV. CLASSES OF COLLECTIVE BEHAVIORA. Class I: Finite Critical Mass (FCM) Models When  ϕ 1  <  0,  γ  1  ≤  0, i.e. in models where  A − B interactions produce on average an increase in undecidedindividuals with no net gain in  A  nor in  B  states, thesystem undergoes a first order transition at a finite value r  =  r c  of the external bias (see Fig. 2, first row). Forlarge enough  r P  1  is the only fixed point and the sys-tem flows into the absorbing state of total consensus onopinion  A  for any initial condition. At  r  =  r c  the systemundergoes a saddle–node bifurcation [23]: two additionalcoinciding fixed points appear. As  r  is decreased be-low  r c  they split (one, with larger  n B , stable, the otherunstable, see Fig. 3(a)). In the nontrivial stable fixedpoint the two opinions  A  and  B  coexist in the popula-tion, together with a fraction of undecided (we call thisstate “pluralism”), see Fig. 4(a). The initial conditionsdetermine whether the pluralistic state ( n B  >  0) or theconsensus state ( n A  = 1) is asymptotically reached. Thevalue of   n B  at the unstable fixed point stays finite in thelimit  r  → 0 (see Fig. 2, first row), implying that a finite“critical mass” [24] of dissenters is always needed to reachthe pluralistic state, no matter how small is the externalbias. B. Class II: Vanishing Critical Mass (VCM)Models This class is identified by  ϕ 1  <  0, 0  < γ  1  < γ  ∗ 1 , andcorresponds to models where  A − B  interactions causeon average a small increase of   B  states at the expenseof   A  states. As in  Class I  , lowering  r  below  r c  thesystem undergoes a first order transition separating aregime ( r > r c ), where consensus on opinion  A  is theonly stable state from a regime ( r < r c ) where a stableand an unstable additional fixed points appear through asaddle–node bifurcation (Fig. 2, second row). However,in this case, further reducing  r , the unstable fixed pointcollides with the point  P  1  (consensus on  A ) at a finitevalue  r ∗ (0  < r ∗ < r c ), and then exits the physical re-gion. This is a transcritical bifurcation [23]: when thetwo fixed points cross each other, they exchange stability(Fig. 3(b));  P  1  becomes unstable, so that below  r ∗ thesystem, unless started with  n B  ≡  0, always flows to thepluralistic state (Fig. 4(b)). The initial presence of even afew dissenters suffices for opinion B  to survive. The curve γ  ∗ 1 ( ϕ 1 ) =  ϕ 1 γ  2 / ( ϕ 1 − ϕ 2 − γ  2 ) separating  Class II   and  III  depends on the parameters  ϕ 2  and  γ  2  (see Appendix B).The transcritical bifurcation line is  r  =  r ∗ = 2 γ  1  (seeAppendix B), and always lies below the saddle–node bi-furcation line (see Fig. 1(b)). C. Class III: Zero Critical Mass (ZCM) Models When  ϕ 1  <  0,  γ  1  ≥ γ  ∗ 1 , corresponding to models where A − B  interactions give an increase of undecided and alarge increase in  B  at the expense of   A , the system un-dergoes at  r  =  r ∗ = 2 γ  1  a continuous transition (trans-critical bifurcation) between total consensus on opinion A  ( r > r ∗ ) and pluralism ( r < r ∗ ), see Fig. 2, third row,and Fig. 4(c). Initial conditions do not play any role. D. Class IV: Total Consensus (TC) Models The region  ϕ 1  ≥ 0 corresponds to models where  A − B interactions result in a net increase of individuals holdingopinion A . The behavior is trivial (see Fig. 3, fourth row):irrespectively of the value of all other parameters, for anyinitial condition, and no matter how small the external  5forcing is, the system always converges to the consensusstate ( P  1 ) (see Fig. 3(c) and 4(d)). V. BEYOND MEAN–FIELD Before discussing the analytical results obtained in theprevious section within MF and examining some of theirconsequences we test their validity beyond MF by nu-merical simulations on synthetic and real–word networks.The simulations are performed on four microscopic dy-namical models, each belonging to one of the universalityclasses derived above.The single event of the dynamics occurs as follows [25].We select randomly a node  i  (speaker) and, with prob-ability  r/ (1 +  r ), we set his state to  A , as effect of theexternal bias. Instead, with complementary probability1 / (1 + r ) an interaction process takes place: we select alistener  j  among the neighbors of   i , and modify the stateof the pair ( i ,  j ) according to Table II in Appendix Bwith probabilities  ψ 2  =  ω 2  =  ϕ 2 ,  δ  2  =  ǫ 2  =  γ  2  and each λ i  =  µ i  =  α i  for any  i . Each single event occurs duringa temporal interval 1 /N   (where  N   is the total number of nodes in the network), so that  N   updates are attemptedin a time unit. Starting from the initial configuration, foreach value of   r  we let the system evolve during 5000 timesteps to reach the stationary configuration and determinethe densities  n A ,  n B  and  n U   by performing averages over5000 additional time steps. In order to characterize thepossible presence of discontinuous phase transitions andthe associated hystereric effects, for each set of data weconsider two different initial conditions: either all nodesin state  B  ( n B  = 1), or all nodes in state A except fora very small fraction of nodes in state  B  ( n A  = 0 . 99, n B  = 0 . 01). We always keep the values  ϕ 2  = 0 . 1 and γ  2  = 0 . 5 fixed and vary  γ  1  and  ϕ 1  in order to encom-pass all four quantitatively distinct behaviors found inthe analytic approach. A. Simulations on synthetic networks In Fig. 5 (upper panel) we plot the stationary valueof the densities as a function of   r  when the interac-tion pattern is given by three synthetic networks of size N   = 20000: a Random Regular Graph (RRG) whereeach node has 10 neighbors; an Erd¨os-R´enyi(ER) graphof average degree 10; a network built using the Uncorre-lated Configuration Model (UCM) with minimum degree3 and degree distribution decaying as  k − 2 . 5 . For all thesecases, predictions are very well matched by numericalsimulations. B. Simulations on real networks A tougher test of the MF results is provided by sim-ulations performed on real–world networks, incorporat-ing additional topological features such as clustering andcorrelations. In Fig. 5 (lower panel) we report resultsfor: a network of size  N   = 81860 representing movieactor collaborations obtained from the Internet MovieDatabase (MOVIES) (average degree   k   = 89 . 53, fluc-tuations  k 2  /  k  = 594 . 91); a network of size  N   = 24608representing connections of Internet Autonomous Sys-tems in 2006 (AS2006) (average degree   k   = 4 . 05, fluc-tuations   k 2  /  k  = 259 . 94) ; the largest connected com-ponent (size  N   = 33696) of the Enron email exchangenetwork (ENRON) (average degree   k  = 10 . 02, fluctua-tions   k 2  /  k  = 140 . 07) [26].Also in these cases numerical simulations agree wellwith the outcome of the MF approach: the behavior ineach of the classes qualitatively reproduces the analyt-ical predictions, with only (expected) variations in theposition of transition points. The only variation in thisrespect concerns class I. In this case it turns out that forvery small values of the rate  r  the state with overwhelm-ing majority of   B  is dynamically reached even startingfrom  n B  as low as 0 . 01, at odds with the MF prediction. VI. DISCUSSION AND CONCLUSIONS We finally discuss some interesting and nontrivial con-sequences that follow from the classification scheme de-rived within the MF approach.Although the parameters regulating the interactionswith undecided agents have a marginal role in determin-ing the collective behavior, the presence itself of the  U  state is crucial in several respects. In the absence of the third, undecided state, the system would either con-verge to total consensus on opinion  A  ( Class IV  )– whenasymmetric interactions favor  A , or exhibit a continuoustransition between consensus and pluralism ( Class III  ) –when asymmetric interactions favor  B  [22].Several models in the literature that allow for a thirdstate require  U   to be a necessary intermediate step whenchanging opinion [21, 27]. Our results imply that suchsystems undergo a discontinuous transition by varyingthe media exposure:  U   being a necessary intermediatestep requires  α 1  =  α 2  = 0, giving  ϕ 1  ≤ 0,  γ  1  ≤ 0; there-fore these models always fall in  Class I  . The condition fora model to be in  Class I   is however more general, onlyexcluding  average   gain of   A  or  B  in  A − B  interactions.Within our framework consensus is always achieved,whatever the interactions, for strong enough media expo-sure. Then a natural question arises: is consensus stableupon removal of the media pressure? The answer to thisquestion is different for different classes. In FCM class,once the consensus is reached, it is kept also when themedia exposure is removed. In ZCM class dissenters nu-cleate as soon as the media exposure is lowered below thethreshold needed to reach consensus ( r < r c ). VCM classshows interesting hysteretic behavior: once consensus isreached with a sufficiently high media exposure ( r > r c ),it is kept when lowering the media pressure that however
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