Majority-vote model with limited visibility: an investigation into filter bubbles
Andre L. M. Vilela, Luiz Felipe C. Pereira, Laercio Dias, H. Eugene Stanley, Luciano R. da Silva
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Majority-vote model with limited visibility: An investigation intofilter bubbles
Andr´e L. M. Vilela,
1, 2, ∗ Luiz Felipe C. Pereira,
3, 4, † LaercioDias, H. Eugene Stanley, and Luciano R. da Silva
3, 5 F´ısica de Materiais, Universidade de Pernambuco, Recife, PE 50720-001, Brazil Center for Polymer Studies and Department of Physics,Boston University, Boston, MA 02215, USA Departamento de F´ısica, Universidade Federal doRio Grande do Norte, Natal, RN 59078-970, Brazil Departamento de F´ısica, Universidade Federalde Pernambuco, Recife, PE 50670-901, Brazil National Institute of Science and Technology for Complex Systems,Centro Brasileiro de Pesquisas Fsicas,Rio de Janeiro, RJ 22290-180, Brazil (Dated: September 2, 2020) bstract The dynamics of opinion formation in a society is a complex phenomenon where many variablesplay an important role. Recently, the influence of algorithms to filter which content is fed to socialnetworks users has come under scrutiny. Supposedly, the algorithms promote marketing strategies,but can also facilitate the formation of filters bubbles in which a user is most likely exposed toopinions that conform to their own. In the two-state majority-vote model an individual adopts anopinion contrary to the majority of its neighbors with probability q , defined as the noise parameter.Here, we introduce a visibility parameter V in the dynamics of the majority-vote model, whichequals the probability of an individual ignoring the opinion of each one of its neighbors. For V = 0 . q c ( V ) and obtain the phase diagram of the model. We find that the critical noise is an increasingfunction of the visibility parameter, such that a lower value of V favors dissensus. Via finite-sizescaling analysis we obtain the critical exponents of the model, which are visibility-independent,and show that the model belongs to the Ising universality class. We compare our results to thecase of a network submitted to a static site dilution, and find that the limited visibility model is amore subtle way of inducing opinion polarization in a social network. PACS numbers: 87.23.Ge Dynamics of social systems, 05.50.+q Lattice theory and statistics, 05.10.LnMonte Carlo methods, 64.60.Cn Order-disorder transformations, 64.60.F- Critical exponentsKeywords: Sociophysics, Phase transition, Critical Phenomena, Monte Carlo simulation, Finite-size scaling ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Opinion formation in a real society is a complex phenomenon where many variables havea marked influence. For simplicity, consider a situation where a population is called uponto choose between two options, such as an election in a bipartisan system or a referendum.Even in this simple scenario, several factors can influence the decision making process ofeach individual. One might consider aspects such as herd mentality and social pressure[1, 2], conformity [3], confirmation bias [4], social dominance [5, 6], and social influence bias[7, 8] among others. Recently, one more aspect has become particularly relevant in opinionformation processes, the influence of individuals and corporations via social networks [9–11].In general, social platforms employ specific algorithms to filter which content is fed to agiven user based on the user’s interactions in the digital world. One of the main objectives ofsuch algorithms is to promote target-oriented marketing strategies. Nonetheless, the samealgorithms can also promote the formation of so-called filters bubbles [12, 13], in which auser is more likely to be exposed to news and opinions which conform to their current beliefs.This kind of filter can isolate users in bubbles, hence the name, where a single opinion isprevalent, even if that opinion is unpopular among the total population. Notice that theformation of filter bubbles depends on the presence of individuals on each others news feed,which is preferred if both have similar views and opinions.Modeling opinion formation using statistical physics methods is not a simple task, even ifthe objective is only a minimally acceptable representation of social reality [14–16]. Nonethe-less, many advances have been made in the field of opinion dynamics, sometimes referredto as Sociophysics [17–19]. Several opinion formation models proposed by physicists arebased on the description employed for magnetic systems, such as the Ising model [14, 20–22]. In this analogy, magnetic moments (spins) become individuals and the exchange energybecomes the social interaction. Therefore, each individual holds one of several possible opin-ions, corresponding to discrete spin states, and the opinion of each individual is influencedby the opinion of its neighbors. The atomic lattice becomes the network of social interac-tions, where each node is occupied by an individual agent, and its topology can take theform of regular or complex networks.Arguably, two of the most popular models in opinion dynamics are the voter model[23, 24] and its generalization, known as the majority-vote model [25]. Originally proposed3y Oliveira, the majority-vote model adds a noise parameter to the voter dynamics, whichcorresponds to the probability of an individual adopting an opinion that differs from themajority of its neighbors [25, 26]. The noise parameter is analogous to the temperature inthe Ising model, and sometimes it is dubbed the social temperature. Starting from zeronoise and increasing its value the system undergoes a phase transition from an orderedphase, where one opinion prevails, to a disordered phase, with no dominant opinion whichcorresponds to a polarized society. The critical noise at which the phase transition takesplace is analogous to the critical temperature in the Ising model.At first, the two-state majority-vote model was investigated on a regular square lattice.It was later generalized to a cubic lattice [27], small-world networks [28, 29], random graphs[30, 31], scale-free networks [32, 33], and spatially embedded networks [34]. The impact ofsite dilution and agent diffusion on the critical behavior of the majority-vote model has alsobeen addressed recently [35, 36], as well as the presence of two types of noises [37]. The modelhas been generalized to three-states and investigated on a square lattice [38–40], randomgraphs [41], and scale-free networks [42]. Further generalizations include a continuous-stateversion of the model [43], the majority-vote model with inertia [44, 45], and the presence ofanticonformists [46] and strong opinions [47]. A variation of the majority-vote model, wherea set of individuals tries to influence the opinion of their neighbouring counterparts has beennamed the block-voter model [48, 49].In this study, we introduce a probabilistic visibility parameter in the majority-vote modelin order to mimic the filter bubble effect. When the visibility is smaller than unit, the indi-vidual will ignore the opinion of some of its neighbors. We perform Monte Carlo simulationsof the two-state majority-vote model on a square lattice to obtain the critical social temper-ature as a function of the visibility, i.e. the phase diagram of the model. Furthermore, weemploy finite-size scaling techniques to estimate the critical exponents which characterizethe phase transition observed in this model. Our results show that the critical noise is anincreasing function of the visibility, and as we lower the visibility a smaller amount of noiseis required to polarize the system. The critical exponents equal those of the Ising model,which places the majority vote model with limited visibility in the Ising universality class,in accordance with the conjecture by Grinstein et al. [50].4
I. LIMITED-VISIBILITY MODEL
The majority-vote model describes a system of interacting individuals where each oneis allowed to be in one two possible states σ i = ±
1. This two-state model evolves in timefollowing a simple update rule: each individual assumes the state of the majority of itsneighbors with probability 1 − q and the opposite state with probability q , independent ofits previous state. It is also possible to express the probability of a given individual changingtheir opinion in terms of their current state and that of the majority of its neighbors [25].The system presents an orderdisorder phase transition as the noise parameter q reaches acritical value q c .The majority-vote model with limited visibility aims to include in the opinion dynamics aparameter that quantifies how each individual i , perceives its neighborhood Λ i or is isolatedfrom it. In this model, we propose that each individual is influenced by the opinion of itsvisible neighborhood Λ ∗ i , governed by a probability V i . The parameter V i is the visibility,defined in the range 0 ≤ V i ≤
1. If V i = 1 the individual considers the opinion of all of itsneighbors equally, while for V i = 0 the individual ignores the opinion of all the neighbors.For 0 < V i <
1, the opinion of each neighbor is considered with probability V i . Hence, in thecase of V i = 1 /
2, the individual will on average consider the opinion of half of its neighbors,ignoring the opinion of the other half. Thus, V i effectively stands for the sight range of agiven individual.In this work, we consider a single visibility parameter for all individuals in the network V i = V for all i . Such dynamics simulate in a simplified way the effect of filters in socialnetworks. Because the algorithm supposedly chooses the content presented to an individualaccording to their reactions to previous content, news from a certain neighbor may be visibleon a given day but not on subsequent days. We model such behavior as a simple randomchoice of which content will be seen at any given moment, abstracting details from the realimplementation of recommendation algorithms, without loss of generality.We define the visibility index I ( V ) as I ( V ) = , with probability V ;0 , with probability 1 − V, (1)where V is the visibility parameter. The dynamics of the model with limited visibility issimilar to that of the standard majority-vote model. The opinion of an individual σ i is5ipped with probability w ( σ i ) = 12 ( − (1 − q ) σ i sgn " k i X δ =1 I ( V ) σ i + δ , (2)where sgn( x ) = +1 , , − x < x = 0, and x >
0, respectively. The sum runs over all k i neighbors connected to the individual i . In our model I ( V ) acts dynamically and it istested for each neighbor. For the special case where the selected individual cannot interactwith any of its neighbors in a given trial, its opinion remains unchanged. In other words, if I ( V ) = 0 for all k i ∈ Λ i (or Λ ∗ i = ∅ ), then w ( σ i ) = 0. Here, we place the individuals on thenodes of a regular two-dimensional square lattice of size N = L × L , such that k i = 4 for all i ∈ N . Naturally, for V = 1 we recover the original majority-vote model [25].Before moving on to the simulation details, we must remark one detail concerning thenature of the visibility parameter. At first sight, one might think that the limited visibilityintroduced in our model is equivalent to a standard site dilution [51, 52]. However, thisis certainly not the case. The dilution is a static feature of the network since an inactivesite will remain dormant during the dynamics of the system. Meanwhile, the visibility ofan individual’s neighborhood proposed here is a fully dynamic feature, and it may changeat each time step. In a sense, the visibility parameter plays the role of a dynamic dilution.Therefore, we do not expect our results to agree with previous studies on statically dilutedsystems. III. SIMULATION DETAILS
In order to investigate the effect of the visibility in the majority-vote model we define anorder parameter, analogous to the magnetization per spin in the Ising model, given by m = 1 L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X i =1 σ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3)We also consider the average of the order parameter given by M ( q, V, L ) = hh m i t i c , (4)where h ... i t stands for time averages taken in the stationary regime, and h ... i c indicatesconfigurational averages taken over independent realizations. The behavior of the model is6urther characterized by the scaled variance of m , which is an extensive quantity analogousto the magnetic susceptibility χ ( q, V, L ) = L (cid:2) hh m i t i c − hh m i t i c (cid:3) , (5)and the kurtosis of m , in the form of the Binder fourth-order cumulant [53] U ( q, V, L ) = 1 − hh m i t i c hh m i t i c . (6)All of the above quantities depend on the noise parameter q , the visibility V , and the systemsize L . Therefore, we expect our model to present a phase transition as q and V are varied,and thus characterize its critical behavior by finite-size scaling analysis [54].In the critical region around the phase transition, the susceptibility presents a maximumat a size-dependent pseudocritical noise parameter, which is related to the true critical noiseby q c ( V, L ) = q c ( V ) + rL − /ν , (7)where r is a numerical constant, and ν is one of the critical exponents. The behavior of M and χ in the vicinity of the phase transition is defined by the following finite-size scalingrelations M ( q, V, L ) = L − β/ν f M ( ǫL /ν ) , (8) χ ( q, V, L ) = L γ/ν e χ ( ǫL /ν ) , (9)where β/ν and γ/ν are critical exponent ratios, and ǫ = q − q c is the distance to the criticalnoise. For each value of V , f M and e χ are universal scaling functions of the variable x = ǫL /ν . The exponent ratios are related to the system dimension by the so-called hyperscalingrelation [54] 2 β/ν + γ/ν = d, (10)which in our case should be satisfied with d = 2, since our model is defined on the squarelattice [42]. The fourth-order cumulant presents its own scaling in terms of a universalfunction U ( q, V, L ) = e U ( ǫL /ν ) . (11)However, unlike M and χ , the cumulant assumes a size-independent universal value at thecritical point e U (0), whose numerical value can be used to identify the university class of thesystem [53, 55]. 7e perform Monte Carlo simulations of the majority-vote model with limited visibilityon square lattices with periodic boundary conditions and linear sizes ranging from L = 8 to200. In our simulations, time is measured in Monte Carlo steps (MCS), which correspondsto N attempts of changing the state of the individuals in the network. Each simulationbegins in a fully ordered state, with all spins set to σ i ( t = 0) = +1 for all i ∈ N , and thus m = 1. For any finite q , the system will take a certain number of MCS to reach its steadystate, so we discard the first 5 × MCS in each simulation. Time averages were calculatedover the next 7 × MCS, and up to 100 independent samples were considered to calculateconfigurational averages.
IV. NUMERICAL RESULTS
Fig. 1 presents the dependence of the average order parameter and the correspondingsusceptibility on the social temperature q , for the majority-vote model with limited visibilityon a square lattice of size L = 200. Each curve corresponds to a visibility parameter rangingfrom 0 .
05 to 1 .
00 (from left to right) with increments ∆ V = 0 .
05. For each value of V ,the system undergoes a phase transition from consensus to dissensus as the noise parameterincreases. The curves show that as the visibility decreases the value of q at which the systembecomes disordered also decreases.In the two-state majority-vote model, consensus is a macroscopic state where one thetwo opinions is adopted by the majority of the society, similar to a ferromagnetic state ofa magnet. The dissensus represents a macroscopic behavior that mimics the paramagneticstate, where the two opinions are distributed symmetrically and M ∼
0. The pseudocriticalnoise values q c ( V, L ) are located at the peaks of the susceptibility, which are in agreementwith the behavior of M ( q, V, L ). From the data in Fig. 1, we conclude that decreasing thevisibility decreases the critical noise required to reach a disordered state. This behaviorindicates that a limited visibility weakens consensus in the system, since a smaller amountof noise is required for the phase transition to take place.In Fig. 2 we fix the visibility V = 0 . q for lattices with sizes L = 40 , , , , ,
140 and 200. Fig. 2(a) shows a phase transition from an orderedstate to a disordered one as the social temperature goes above some critical value of q . In8 IG. 1: (Color online) The social temperature dependence of (a) order parameter and (b) suscep-tibility for the majority-vote model with limited visibility for L = 200. The visibility parameterincreases from left V = 0 .
05 to right V = 1 with ∆ V = 0 .
05. Error bars are smaller than thesymbol size. Lines are guides to the eye.
Fig. 2(b), the susceptibility reaches a maximum in the critical region q ≃ q c ( V, L ), whichbecome sharper for larger system sizes. In Fig. 2(c) the cumulant for systems of differentsize intersect at the critical point q c ( V = 0 .
8) = 0 . q compared to the isotropic case, since q c ( V = 1) = 0 . q c on V is shown in Fig. 3, where we present the phase diagram of themajority vote model with limited visibility. In this plot, we consider the critical points q c ( V )estimated from the crossing of the Binder cumulant for each V . The circles represent thenumerical values and the error bars are smaller than the symbols. The critical points separate9 IG. 2: (Color online) (a) Order parameter, (b) susceptibility, and (c) re-scaled fourth-ordercumulant as a function of the noise parameter for several values of the system size L and fixedvisibility V = 0 .
8. Within the accuracy of the data, all curves in (c) intersect at q c = 0 . an ordered phase, for which the system presents consensus, from a disordered phase, wheredissensus dominates. Once again, we observe that the lower the visibility of an individual’sneighborhood, the lower the critical noise that drives the system to the disordered state.Thus, the critical social temperature is an increasing function of the visibility parameter,and limiting the visibility can effectively hinder the formation of an ordered state.Table I lists the values of the critical noise for several values of the visibility parameter V .The table also presents results for the static dilution case which will be discussed later. Theresults for q c ( V ) indicate that the control of the visibility is the key to promote or suppresspolarization in a society. Therefore, bubble filters have a negative impact in the formationof consensus in a population, since individuals are more likely to ignore opinions oppositeto their own. We note that for V < .
5, the amount of social noise required to destroyconsensus is lower than 0 .
01. Therefore, if we consider a social network where individualsagree with only half of its connections, even the smallest amount of noise (or disagreement)is capable of originating a polarized society. In what follows, we focus our presentation fornetworks where V ≥ . M and (b) ln χ at the critical point q = q c ( V )versus ln L for V = 1 . , . , . , . , . .
5. The angular coefficient of the straight lines,obtained from a linear regression to the simulation data, confirm the scaling of M and χ IG. 3: (Color online) Phase diagram of the majority-vote model on a square lattice, obtainedfrom the crossing of the Binder cumulant. Circles represent the critical noise for the model withlimited visibility, and squares are for the static site dilution case. Error bars are smaller than thesymbols. In the green (white) region, the system presents consensus (disensus) for the model withlimited visibility. The the red line is a polynomial fit to the data q c ( V ) = a + bV + cV , where a = − . b = 0 . c = − . according to Eqs. (8) and (9). In Fig. 4(c) we plot ln[ q c ( V, L ) − q c ( V )] versus ln L , fromwhich we obtain the critical exponent 1 /ν according to Eq. (7). The slope of the lines forFig. 4 (a), (b) and (c) correspond to critical exponent ratios β/ν = 0 . γ/ν = 1 .
75 and1 /ν = 1 .
0. Those values indicate that the majority-vote model on a square lattice withlimited visibility belongs to the Ising universality class. This finding is further corroboratedby the size-independent universal value calculated for the Binder cumulant e U (0) = 0 . β/ν + γ/ν = d is satisfied with d = 2, asone would expect for the square lattice [42, 54].Using the finite size scaling relations from Eqs. (8)-(7) with the critical exponents ofthe model it is possible to obtain the universal functions f M ( x ), e χ ( x ) and e U ( x ) shown11 ABLE I: The critical noise as a function of the visibility parameter for our model q c ( V ) and forthe static dilution case. V q c ( V ) q static dilution c . . . . . . . . . . . . . . − . . − . . − . . − . . − . . − in Fig. 5 versus the scaled variable x = ǫL /ν . The curves are shifted down to avoidoverlapping between them, but yield only one size-independent universal curve for f M and e χ , for each visibility. We obtain similar results for the scaled M , χ and U for all V ∈ (0 , V. LIMITED VISIBILITY VERSUS STATIC DILUTION
In Fig. 3 and Tab. I we also show the critical noise values for the majority-vote modelwith a static dilution (red squares in the phase diagram). In this case, for each realization,we select a fraction V of nodes that remain present in the network, and permanently removethe remaining 1 − V fraction. For the sake of consistency with the limited visibility model,we enforce that an individual remains in its original state if there are no neighbors connectedto it. The solid red line in Fig. 3 was obtained from a polynomial fit to the data in theform q c ( V ) = a + bV + cV , where a = − . b = 0 . c = − . IG. 4: (Color online) Plot of (a) ln M ( q c , V, L ), (b) ln χ ( q c , V, L ) and ln[ q c ( V, L ) − q c ( V )] versusln L . From top to bottom, V = 1 . , . , . , . , . .
5. Straight lines are obtained from alinear regression to the data, and their slope equals the respective critical exponents. Taking theerror bars into consideration, we find (a) β/ν = 0 . γ/ν = 1 .
75, and (c) 1 /ν = 1 . result agrees well with previous investigations on the majority-vote dynamics with dilution[35, 36].The simulation results show that the critical noise required to reach the disordered stateis lower for the static dilution when compared with the dynamic dilution of the limited13 IG. 5: (Color online) Scaling plot of the (a) order parameter, (b) susceptibility and (c) fourth-order cumulant for lattice sizes L = 60 , ,
100 and 120. From top to bottom we have V =0 . , . , . , . , . .
0. For all scaling plots, we use β/ν = 0 . γ/ν = 1 .
75 and 1 /ν = 1 . visibility model. We conclude that the static dilution and the limited visibility model areboth capable of promoting polarization, yet via different mechanisms. While the staticdilution improves order undergoing an abrupt interference, where individuals may noticethe permanent absence of their neighboring nodes, the dynamic dilution promotes orderingby a more subtle approach. In the latter, one individual will most likely interact with anever-changing reduced neighborhood at any given time, instead of interacting exclusivelywith the same individuals. We remark that the limited visibility model also presents abroader tuning range for the control parameter V , where one can improve the ordering byadjusting this value smoothly and continuously. VI. CONCLUSION AND FINAL REMARKS
In this work, we introduced a visibility parameter in the two-state majority-vote modelin order to capture the possible effects of filter bubbles on opinion formation dynamics. Thevisibility parameter, 0 ≤ V ≤
1, equals the probability of an individual considering theopinion of one of its neighbors when determining the majority opinion. When the visibilityequals unit we recover the original model. The geometric structure of social interactionssupports an order-disorder phase transition when the social temperature is increased abovea critical value, which is an increasing function of the visibility parameter. Our investigationshows that one might promote or suppress polarization in a social network by controlling14he visible neighborhood of an individual, in other words, controlling the effective influenceof its closest social group.Numerical results exhibit the typical finite-size effects for the order parameter when thesystem undergoes a second order phase transition. We obtain the critical exponents of themodel via finite-size scaling to find β/ν = 0 . γ/ν = 1 .
75 and 1 /ν = 1 .
0. Moreover, usingthe hyperscaling relation we obtain d = 2 as expected for a two-dimensional square latticeof social interactions. The calculated critical exponents do not depend on the visibilityparameter and place the majority-vote model with limited visibility in the Ising universalityclass.Our investigation covers a key feature on how to promote or suppress polarization in asocial network. We remark that preventing consensus by means of a limited visibility mightalso be implemented to prevent negative effects, such as the spread of fake news. A naturalextension of the limited-visibility opinion dynamics include the use of complex networksto map the geometry of social interactions. We also avail that the use of different proba-bility distributions for the visibility index I ( V ) might produce exuberant phase transitionphenomena. Acknowledgements
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