Polarization inhibits the phase transition of Axelrod's model
Carlos Gracia-Lázaro, Edgardo Brigatti, Alexis R. Hernández, Yamir Moreno
PPolarization inhibits the phase transition of Axelrod’s model
Carlos Gracia-L´azaro, Edgardo Brigatti, Alexis R. Hern´andez, and Yamir Moreno
1, 3, 4 Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI),Universidad de Zaragoza, 50018 Zaragoza, Spain Instituto de F´ısica, Universidade Federal do Rio de Janeiro, 22452-970 Rio de Janeiro, Brazil Departamento de F´ısica Te´orica. Universidad de Zaragoza, Zaragoza E-50009, Spain ISI Foundation, Turin, Italy (Dated: February 16, 2021)We study the effect of polarization in Axelrod’s model of cultural dissemination. This is donethrough the introduction of a cultural feature that takes only two values, while the other featurescan present a larger number of possible traits. Our numerical results and mean-field approximationsshow that polarization reduces the characteristic phase transition of the original model to a finite-size effect, since at the thermodynamic limit only the ordered phase is present. Furthermore, forfinite system sizes, the stationary state depends on the percolation threshold of the network wherethe model is implemented: a polarized phase is obtained for percolation thresholds below 1/2, afragmented multicultural one otherwise.
I. INTRODUCTION
Agent-based models (ABM) [1, 2] provide a fruitfultheoretical framework to study the fundamental mech-anisms underlying the dynamics of social systems [3].In this line, the Axelrod’s Model for cultural dissemina-tion [4], introduced in 1977 by Robert Axelrod, has be-come a paradigm for the study of social imitation. Themodel relays on the idea of homophily, i.e. , agents in-teract more likely with similar neighbors, and therefore,similar neighbors tend to become even more alike. Toimplement this idea in the model, the probability for anagent to imitate a neighbor’s uncommon cultural traitis proportional to the number of other traits that bothalready share. Axelrod found that, while for low val-ues of the initial cultural diversity the dynamics drivesthe system towards a monocultural state, for larger ini-tial diversity the system freezes in a multicultural state.The Axelrod’s Model has been studied under differentapproaches and variations, including complex networks[5, 6], clustering [7], social pressure [8], noise [9, 10], ex-ternal fields [11, 12], dynamic features [13], mobility andsegregation [14, 15], tolerance [16], confidence thresholds[17], and dynamic networks [18].In this work, we are interested in adding to the Ax-elrod’s model an element that can account for polariza-tion. Cultural, ideological, and political polarizations arephenomena that recently have attracted the attention ofthe scientific community, politics, and society as a whole[19–22]. A source of this renewed interest stems from theobservation that, paradoxically, polarization may consti-tute a side effect of globalization. Although accordingto the homogenization thesis, globalization should stan-dardize a global pattern [23–25], cultural alternatives andresistance to Western norms suggest that culture univer-salization may lead to polarization [26, 27]. These twotheses constitute the convergence-divergence open debate[28]. To this end, by introducing a cultural feature withonly two possible values ( e.g. , left-right, east-west), wepropose a modification of the Axelrod’s model that al- lows exploring the consequences of polarizing issues oncultural diversity.The introduction of a polarizing issue in the Axelrod’smodel seems to have an important impact on the exis-tence of its phase transition. The order-disorder transi-tion described by Castellano et al. [29] is a genuine phasetransition which occurs in the limit of infinite system size[30, 31]. Anyway, it has already been questioned in rela-tion to its robustness. In some circumstances, it has beenproven that exogenous perturbations can drive the sys-tem to a monocultural state [10]. Another interesting ex-ample is the effect that cultural drift, modeled as a noisewhich randomly changes one agent’s cultural trait, hason the model. In this case, even if a finite system showsa well defined noise value which separates the transitionbetween order and disorder, an infinite system always re-sults in a multicultural state [9, 10]. The disappearanceof the transition is also recorded when the original Axel-rod’s model is embedded on a Barab´asi-Albert network[5]. In that case, the location of the finite-size transitionpoint scales as a power of the system size with a positiveexponent, and, therefore, in the thermodynamic limit,the transition disappears because the ordered monocul-tural state always establishes in the system. Keeping inmind this interesting aspects of the Axelrod’s model, theprincipal aim of this work is to analyze the effects of theintroduction of cultural polarization on its phase transi-tion and to characterize the impact of this new elementon the dynamics of cultural dissemination.
II. MODEL
The original Axelrod’s model of cultural disseminationconsiders N agents interconnected by a network whoselinks represent the social interactions. For an agent i ,the culture is represented by a set of F variables { σ f ( i ) } ( f = 1 , ..., F ), the cultural features , that can assume q values, σ f = 0 , , ...q −
1, the traits of the feature. Thenumber of possible traits, q , represents the initial cultural a r X i v : . [ phy s i c s . s o c - ph ] F e b diversity, which is obtained by means of an equiprobablerandom initialization of each agent’s features. At eachtime step, an agent i is chosen at random and allowed toimitate an uncommon feature’s trait of a randomly cho-sen neighbor j with a probability given by their culturaloverlap ω ij , which is defined as the fraction of commoncultural features: ω ij = 1 F F (cid:88) f =1 δ σ f ( i ) ,σ f ( j ) , (1)here δ x,y stands for the Kronecker’s delta, defined as δ xy = 1 if x = y and δ xy = 0 otherwise.In this work, we will consider the situation where oneof the F features, f = 1, is limited to take only two val-ues ( σ ( i ) ∈ { , } , ∀ i ∈ { , , . . . , N } ). The rest of thefeatures, f = 2 , , . . . , F , can take q possible values, as inthe original model. In this way, we will address how a di-chotomy can impact the process of cultural disseminationdescribed by the Axelrod’s model. III. RESULTS
The principal aim of our study is the characterizationof the phase transition exhibited by the model. To thisend, we simulate the model on a regular two-dimensionalsquare lattice of size L ( i.e. , N = L nodes, each node be-ing occupied by a cultural agent) with periodic boundaryconditions. We consider von Neumann neighborhoods sothat each agent has k = 4 neighbors, and interactionstake place only between two neighbors. The number offeatures is fixed to F = 10 and the parameters q and N are varied.The phase transition is determined by the passagefrom an absorbing monocultural state composed by asingle cultural cluster where, for small q values, every-body shares the same culture to a frozen disordered andfragmented state. In fact, for large values of q order isnot attained and different cultures, distributed amongthe sites, characterize the system.Because of this phenomenology, the relative size ofthe largest cultural cluster present in the system isan excellent parameter for characterizing the transition[29, 32, 33]. This parameter is defined as the size ofthe largest cluster, made up by nodes sharing all thetraits, normalized by the system size: S max /N . This pa-rameter is estimated averaging over different simulations: (cid:104) S max /N (cid:105) .Panel A of Fig. 1 shows the order parameter (cid:104) S max /N (cid:105) versus the initial cultural diversity q for different systemsizes N . The inspection of the behavior of the orderparameter suggests the existence of a transition. In fact,a sharp transition, characterized by a drop of (cid:104) S max /N (cid:105) for a critical value of q , is observed. We can identify thistransition point looking at the fluctuations of the size ofthe largest cluster: q < S m a x / N > N=900N=1600N=2500N=10000N=14000 q / N < S m a x / N > N=900N=1600N=2500N=10000N=14000 q N=100N=400N=900N=1600N=2500N=10000N=12100 N × × × × × × × q c S max <1 max Var (cid:114) q / N N=100N=400N=900N=1600N=2500N=10000N=12100 (cid:114)
A BC D
Figure 1: A : Mean of the normalized largest cultural clustersize (cid:104) S max /N (cid:105) as a function of the initial cultural diversity q for different system sizes N . B : The same plot after rescalingthe x-axis for q/N . C : Variance χ of the order parameter (cid:104) S max /N (cid:105) versus q for different system sizes. In the inset:Value of q c for the maximum variance χ (blue squares) andfor the jump location in the normalized size of the largestcluster (red Xs) as a function of the system size N , showingthe tendency of the finite-size transition points q c ( N ) in thethermodynamic limit: q c = lim N →∞ q c ( N ) = ∞ . D : χ versus q/N for different system sizes N . It is shown that the max-imum of the fluctuations in the order parameter takes placefor the same value of q/N regardless of the system size. Allthese results correspond to a lattice (k=4) and F=10. Eachpoint is averaged over 1000 simulations. χ = (cid:104) S max (cid:105) − (cid:104) S max (cid:105) . Panel C of Fig. 1 shows the values of χ as a functionof q for different system sizes N . This quantity displaysits maximum at a value of q that can be considered asthe finite-size transition point q c ( N ). From this analy-sis we can estimate the critical point by looking at theconvergence of the finite-size transition points q c ( N ), asestimated by the localization of the maxima of the fluc-tuations. The inset of panel C shows that the finite-sizetransition point values grow proportionally to the sys-tem size as: q c ( N ) ∝ N . The same conclusion is ob-tained measuring the transition points from the locationof the observed discontinuity in the normalized size ofthe largest cluster. This point corresponds to the firstvalue of q for which (cid:104) S max /N (cid:105) is less than 1. Using theseresults, we can obtain a data collapse by introducing therescaled parameter q/N (see panels B and D of Fig. 1).This scaling indicates that, in the thermodynamiclimit, the transition point ( q c = lim N →∞ q c ( N )) goes toinfinite. Therefore, the system does not display a gen-uine phase transition, which is rigorously defined at thethermodynamic limit, where the number of constituentstends to infinity. The introduction of the binary featuredestroys the well known phase transition of the Axel-rod’s model: in the thermodynamic limit the transitiondisappears and the ordered monocultural state is the solephase displayed by the system. t BA t ρ ρ Figure 2: Evolution of the density ρ of active links in aregular lattice ( N = 1600, k = 4) for q = 100 ( A ), q =5 × ( B ). Each color corresponds to a realization amongten randomly selected. A clear understanding of this result can come from theexploration of the dynamics of the density ρ of activelinks. An active link is a bond that connects two agents( i, j ) with at least one different feature and at least an-other one equal ( i.e. , 0 < ω ij < ρ = 0 implies a frozen configuration. Figure 2 dis-plays the time evolution of this quantity. Panel A ofFigure 2 shows the result of ten different simulations for q < q c ( N ), which generate ordered final configurations.In this situation, ρ starts near 0.5, experiences a decreasebut then rises again towards a peak before finally decay-ing to zero. The second part of the dynamics, after themaximum is reached, is a coarsening process which fol-lows different erratic paths and abruptly reaches zero.Note that the final state corresponding to one region ofordered features is reached as the result of a fluctuationin a finite system. A finite size effect produces a finalordered region of size comparable to the whole system.When q > q c ( N ) (Panel B of Figure 2) the simulationsdo not converge to an ordered state and ρ follows quiteregular paths towards zero. This smooth coarsening pro-cess gives rise to regions clearly smaller than the systemsize (of the order of N/
10) and produces regular similartrajectories of ρ ( t ). These results are totally analogousto those of the original Axelrod’s model [29].In order to better understand the impact of the binaryfeature on the evolution of the system, we have studiedthe dynamics of the mean overlap (cid:104) ω ( t ) (cid:105) . Panels A - C of Figure 3 display the trajectories of the mean over-lap of the binary feature ( (cid:104) ω B ( t ) (cid:105) ) versus the mean over-lap of the non-binary features ( (cid:104) ω NB ( t ) (cid:105) ). Solid lines NB ω ω < B > < > < NB > < B > AB < NB > < B > C ωω ωω Figure 3: Evolution of the mean overlap for the binary featureversus the overlap for the non-binary features. Solid linescorrespond to the numerical results for a regular lattice ( k =4, N = 1600) and q = 10 ( A ), 10 ( B ), 10 ( C ). Differentcolors correspond to 10 different characteristic realizations.Dashed blue lines show the results corresponding to the mean-field approximation given in Eq. (2). correspond to the numerical results, different colors cor-responding to different single simulations. We can dis-tinguish three different stages. During the first stage,the binary feature remains almost unaltered, in contrastto the non-binary overlap which increases. This sug-gests that the cultural uniformization is realized pre-dominantly within clusters of agents that share the bi-nary trait. For q > q c ( N ) (Panel C ), only this firststage takes place, and the system remains frozen in adisordered state with a characteristic number of differ-ent cultural clusters. For q (cid:46) q c ( N ) (Panels A - B ), aftera particular overlap value of the non-binary feature isreached, the two overlaps start to increase together. Inthis regime, the cultural exchange might take place alsobetween agents marked by distinct binary features. Fi-nally, a finite-size driven process turns on and the systemconverges to the ordered phase. It is interesting to notethat, in general, the binary feature reaches the conver-gence before the non-binary ones. m,sm-1,sm,s=0m,s n,s’=0m,s n,s’ m+1,s or m,s+1m,sm,s=1m,s n,s’=1m-1,s n+1,s’ a)b)c)d)e) m,s n,s’m-1,s n,s’m+1,s n,s’ m,s n+1,s’m,s n+1,s’m+1,s n+1,s’m,s n,s’=0 m,s n,s’=1 f)g)h)i) Figure 4: Schematic description of the different processes thatcontribute to equation 2. a ) to c ) correspond to direct pro-cesses while d ) to i ) correspond to indirect ones. To illus-trate the meaning of these diagrams lets consider two of them.The sub-diagram b ) corresponds to the direct process where a( m − , s ) link evolves to a ( m, s ) link. The probability for thisprocess to happen is given, in a mean-field approximation, bythe probability of a ( m − , s ) link to be sorted ( P m − ,s ),times, the probability of the imitation process to happen( m − s/F ), times, the probability for the imitation to occuron a non-binary feature (( F − m + s ) / ( F − m +1)), which pro-duces the second term of equation (2). The sub-diagram e )corresponds to the indirect process where the state change ofa link, from ( n, s (cid:48) ) to ( n +1 , s (cid:48) ), also modifies one of the ( k − m, s ) to ( m − , s ). The probability forthis process to occur is approximated by the probability of a( n, s (cid:48) ) link to be sorted ( P n,s (cid:48) ), times, the probability to havea ( m, s ) as a neighbor link (( k − · P m,s ), times, the probabil-ity for the imitation to happen (( n + s (cid:48) ) /F ), times, the proba-bility to imitate a non-binary feature (( F − − n ) / ( F − n − s (cid:48) )),times, the probability that the chosen feature was one of the m shared features ( m/ ( F − To gain a better understanding of the overlaps dynam-ics, here we extend the single link mean-field approachdescribed in [29] to the case where one of the features isrestricted to only two values. In order to do that, we de-fine P m,s as the probability for two neighbors to coincidein m features (out of F −
1) and s = 0 , P m,s . To compute theseprobabilities, we notice that when an agent j imitates afeature from an agent z , it modifies the state of the linkbetween j and z (direct process) and the state of someof the other links of agent j (indirect processes). Figure4 displays a diagrammatic description of all the possibleprocesses that contribute to the P m,s dynamics.By considering all the depicted diagrams we finally ar- rive to: dP m,s dt = − P m,s m + sF (1 − δ F,m δ s, ) + P m − ,s m − sF (cid:18) F − m + sF − m + 1 (cid:19) + P m,s − mF (cid:18) sF − m (cid:19) +( k − × − (cid:88) n,s (cid:48) P m,s P n,s (cid:48) n + s (cid:48) F − s (cid:48) F − n − s (cid:48) − (cid:88) n,s (cid:48) P m,s P n,s (cid:48) n + s (cid:48) F F − − nF − n − s (cid:48) mF − − (cid:88) n,s (cid:48) P m,s P n,s (cid:48) n + s (cid:48) F F − − nF − n − s (cid:48) F − − mF − ω δ ( s,s (cid:48) ) ( t )+ (cid:88) n,s (cid:48) P m − ,s P n,s (cid:48) n + s (cid:48) F F − n − F − n − s (cid:48) F − mF − ω δ ( s,s (cid:48) ) ( t )+ (cid:88) n,s (cid:48) P m, ¯ s P n,s (cid:48) nF − s (cid:48) F − n − s (cid:48) + (cid:88) n,s (cid:48) P m +1 ,s P n,s (cid:48) n + s (cid:48) F F − n − F − n − s (cid:48) m + 1 F − , (2)where k is the mean connectivity of the network, δ ( a, b )is the Kronecker’s delta function, ¯ s stands for the logicnegation of s , and ω ( t ) ( resp ., ω ( t )) is the mean overlapbetween agents that share (not share) the binary trait.We also consider P − ,s = P F +1 ,s = 0.Dashed blue lines in panels A - C of Figure 3 show theevolution of the binary feature’s overlap versus the non-binary features overlap, according to the mean-field ap-proximation. As shown, the mean-field approximationqualitatively reproduces the behavior of all the regimes,including the first two until the finite size of the numeri-cal simulations allows the final convergence for q (cid:46) q c .To test if the mean-field description can reproduce theeffect of the connectivity on the dynamical behavior, werun some simulations of our model implemented on a ran-dom regular network (RRN) with different values of thedegree k . The thin solid lines in Figure 5 display thenumerical results for the evolution of binary and non-binary overlaps for the RRN with increasing connectiv-ity: k = 4 , , A , B , C , respectively. The thickdashed lines correspond to the mean-field approximationfor the same values of k . As shown, the mean-field ap-proach describes the decreasing value of the non-binaryoverlap reached after the first part of the dynamics, be-fore the finite-size driven coarsening dynamics start.As mean-field descriptions assume an infinite systemsize, the considered approximation can be useful also forshedding light on the absence of the transition in thethermodynamic limit, corroborating our numerical find-ings with some analytic arguments. If we consider a sys-tem of infinite size, as already suggested by Castellano etal. [29], for q < q c the system is posed indefinitely in acoarse-grained state and, for q > q c , after a characteris-tic time, the coarsening process stops and the density ofactive links ρ equals zero. Hence, for infinite systems, wecan rigorously define the transition looking at the valueof ρ , which acts as an order parameter distinguishingamong the two different dynamical regimes: one with aperennial coarsening state ( ρ > ρ = 0). Therefore, if a phase transition exists ρ undergoes a discontinuity, jumping from a finite valueto zero. In our mean-field approximation ρ can be com-puted by solving the system of equations 2, which leadto: ρ = (cid:80) F − f =1 P f,s =0 + (cid:80) F − f =0 P f,s =1 . The correspond-ing stationary values of ρ are independent of q and al-ways positive, without any discontinuity, implying thatit is not possible to detect any phase transition in theinfinite-size limit.Finally, we study the model behavior in the regimethat does not lead to a monocultural state ( q > q c ).When the site percolation threshold of the network is be-low 1 /
2, agents sharing a given binary trait form a giantcomponent even for a random distribution of the traits.Therefore, initially, there will be two binary-feature clus-ters, one for each binary trait, allowing cultural imita-tion inside those components, since any pair of connectednodes belonging to one of those components has a non-zero cultural overlap. Those two clusters will convergeseparately, leading the system to a final polarized state.Conversely, when the percolation threshold of the net-work is above 1 /
2, initially there are no giant componentsfor the binary traits and, presumably, for large enoughvalues of q and finite system sizes, initial small clustersof agents sharing the binary feature will converge lead-ing the system to a frozen cultural mosaic constitutinga multicultural fragmented phase [14, 16]. Summarizing,for large values of q , the final state can correspond ei-ther to a fragmented phase with a large number (whichscales with the system size) of different cultures or to apolarized state, where the number of different cultures isfinite, with two predominant ones.To test this hypothesis, we have implemented themodel in both a regular square lattice with connectiv-ity k = 4, which presents a site percolation thresholdabove 1 / . k = 4, whosesite percolation threshold is below 1 / . S/N ) of the two largest cultures for bothnetworks. As shown in Panel A , for q > q c ( N ), in a reg-ular lattice, the two more spread cultures cover around10% of the system size. In the case of the RRN (Panel B ), for the same q values, the two more spread commu- < NB>00.20.40.60.81 < B > A < NB > < B > B ω ω < B > C ω ω ω ω < NB > Figure 5: Solid lines show the evolution of the overlap for thebinary feature versus the overlap for the non-binary featuresaccording to simulations run on a RRN ( N = 1600) with k = 4 ( A ), 6 ( B ), 8 ( C ); different colors correspond to 10 dif-ferent characteristic runs. Dashed blue lines show the resultscorresponding to the mean-field approximation. q = 10 . nities include near 90% of the system, and the remainingsites correspond to a pool of different cultures organizedon the border separating the two main clusters. IV. DISCUSSION AND CONCLUSIONS
In summary, we have presented a study of a modifiedversion of Axelrod’s model to describe the effect of cul-
50 60 q / N < S / N > st culture2 nd culture A st culture2 nd culture < S / N > q / N B Figure 6: Normalized size of the communities correspondingto the two largest cultures as a function of q for the modelimplemented on a regular square lattice (Panel A ) and on aRRN ( B ), both with connectivity k = 4. Averages are takenover 100 independent simulations and N = 1600. tural polarization in a population of interacting agents.This characteristic is introduced by limiting one of thecultural features to take only two possible values. Themodel was first analyzed on a regular 2D square latticewith the intention of clearly characterizing the nature ofthe transition between the ordered monocultural stateand the disordered one. Our analysis shows that the in-troduction of a binary feature makes the well-establishedphase transition of the classical Axelrod’s model to dis-appear in the thermodynamic limit. This behavior hasbeen characterized through a finite-size scaling analysisbased on the variance of the order parameter.These results show that the system does not display agenuine phase transition. Even if this point is not a gen-uine critical point, it has a clear physical meaning as it isthe value of the q parameter that identifies the shift fromthe order to the disorder regime. Other models, wherethe transition is only observed for finite size systems, dis-appearing in the thermodynamic limit, and which presentsystem size scaling, are well known in the literature [34–37]. In particular, this phenomenon has been previouslyobserved in the Axelrod’s model [5, 9, 10, 14, 16].In general, when we transfer statistical physics toolsto problems of social sciences, the population size is al-ways considerably smaller than the Avogadro number,and so, not so large to justify the thermodynamic limitand its results. In fact, we are interested in the behaviorof finite-size systems, where important phenomena canappear regardless of the number of individuals [34]. Forthis reason, it is interesting to characterize the behaviorof our system for typical finite numbers of individuals.We have presented some results for the overlap dynam- ics of both binary and not binary features, displayingthree different stages. In the first stage, the cultural uni-formization takes place predominantly inside groups ofagents with the same binary trait. Then, only for low q values, a new regime of coarsening takes place, with afirst step where the cultural exchange also happens be-tween agents, marked by distinct binary features and,finally, a finite-size driven process that leads the systemto converge to the ordered phase. This behavior suggeststhat the binary feature has a predominant role in char-acterizing the dynamics of the system. This fact can beinterpreted in social terms: agents find empathy on thebase of some polarizing feature and then tend to aligntheir cultures. Consequently, the spatial structure in theinitial condition of the binary feature clusters is essen-tial to characterize the dynamics of the system. Varyingthese initial conditions, it is possible to modify the dy-namics and to pose the system in different disorderedabsorbing states. This is possible by changing the con-nectivity of the network of interactions. We consideredthe case of a regular square lattice and random regularnetworks with different average degrees. For the regu-lar lattice, which presents a percolation threshold largerthan 0 .
5, there is no giant component in the cluster ofthe binary feature and the disordered absorbing state ofthe system is a fragmented one. For the random regularnetworks, which present a percolation threshold smallerthan 0 .
5, the clusters of the binary feature are giant com-ponents, and, therefore, the disordered absorbing stateof the system corresponds to cultural polarization. Incontrast to the original Axelrod’s model, always char-acterized by fragmentation and with a large number ofdifferent cultures scaling with the system size, our sys-tem can be partitioned in a few different cultures. This isan interesting result because models with a large numberof possible different absorbing states that present polar-ization are uncommon [32, 38]. We conclude that ourimplementation can effectively describe polarization infinite populations.
Acknowledgments
C. G. L. and Y. M. acknowledge partial support fromProject No. UZ-I-2015/022/PIP, the Government ofArag´on, Spain, FEDER Funds, through Grant No. E36-17R to FENOL, and from MINECO and FEDER funds(Grant No. FIS2017-87519-P). [1] M. W. Macy and R. Willer, “From factors to factors:computational sociology and agent-based modeling,”
An-nual review of sociology , vol. 28, no. 1, pp. 143–166, 2002.[2] L. Tesfatsion and K. L. Judd,
Handbook of computationaleconomics: agent-based computational economics , vol. 2. Elsevier, 2006.[3] C. Castellano, S. Fortunato, and V. Loreto, “Statisticalphysics of social dynamics,”
Reviews of modern physics ,vol. 81, no. 2, p. 591, 2009.[4] R. Axelrod, “The dissemination of culture a model with local convergence and global polarization,”
Journal ofconflict resolution , vol. 41, no. 2, pp. 203–226, 1997.[5] K. Klemm, V. M. Egu´ıluz, R. Toral, and M. San Miguel,“Nonequilibrium transitions in complex networks: Amodel of social interaction,”
Physical Review E , vol. 67,no. 2, p. 026120, 2003.[6] B. Guerra, J. Poncela, J. G´omez-Garde˜nes, V. Latora,and Y. Moreno, “Dynamical organization towards con-sensus in the axelrod model on complex networks,”
Phys-ical Review E , vol. 81, no. 5, p. 056105, 2010.[7] N. Lanchier et al. , “The axelrod model for the dissemi-nation of culture revisited,”
The Annals of Applied Prob-ability , vol. 22, no. 2, pp. 860–880, 2012.[8] M. Kuperman, “Cultural propagation on social net-works,”
Physical Review E , vol. 73, no. 4, p. 046139, 2006.[9] K. Klemm, V. M. Egu´ıluz, R. Toral, and M. San Miguel,“Global culture: A noise-induced transition in finite sys-tems,”
Physical Review E , vol. 67, no. 4, p. 045101, 2003.[10] K. Klemm, V. M. Egu´ıluz, R. Toral, and M. San Miguel,“Globalization, polarization and cultural drift,”
Jour-nal of Economic Dynamics and Control , vol. 29, no. 1,pp. 321–334, 2005.[11] J. C. Gonz´alez-Avella, M. G. Cosenza, and K. Tucci,“Nonequilibrium transition induced by mass media in amodel for social influence,”
Physical Review E , vol. 72,no. 6, p. 065102, 2005.[12] J. C. Gonz´alez-Avella, M. G. Cosenza, K. Klemm, V. M.Egu´ıluz, and M. S. Miguel, “Information feedback andmass media effects in cultural dynamics,” arXiv preprintarXiv:0705.1091 , 2007.[13] A. R. Hern´andez, C. Gracia-L´azaro, E. Brigatti, andY. Moreno, “Robustness of cultural communities in anopen-ended axelrod’s model,”
Physica A: Statistical Me-chanics and Its Applications , vol. 509, pp. 492–500, 2018.[14] C. Gracia-L´azaro, L. F. Lafuerza, L. M. Flor´ıa, andY. Moreno, “Residential segregation and cultural dissem-ination: An axelrod-schelling model,”
Physical Review E ,vol. 80, no. 4, p. 046123, 2009.[15] J. Pfau, M. Kirley, and Y. Kashima, “The co-evolutionof cultures, social network communities, and agent lo-cations in an extension of axelrod’s model of culturaldissemination,”
Physica A: Statistical Mechanics and itsApplications , vol. 392, no. 2, pp. 381–391, 2013.[16] C. Gracia-L´azaro, L. Floria, and Y. Moreno, “Selec-tive advantage of tolerant cultural traits in the axelrod-schelling model,”
Physical Review E , vol. 83, no. 5,p. 056103, 2011.[17] L. De Sanctis and T. Galla, “Effects of noise and confi-dence thresholds in nominal and metric axelrod dynam-ics of social influence,”
Physical Review E , vol. 79, no. 4,p. 046108, 2009.[18] C. Gracia-L´azaro, F. Quijandr´ıa, L. Hern´andez, L. M.Flor´ıa, and Y. Moreno, “Coevolutionary network ap-proach to cultural dynamics controlled by intolerance,”
Physical Review E , vol. 84, no. 6, p. 067101, 2011.[19] M. D. Conover, J. Ratkiewicz, M. Francisco,B. Gon¸calves, F. Menczer, and A. Flammini, “Po-litical polarization on twitter,” in
Fifth internationalAAAI conference on weblogs and social media , 2011.[20] M. Prior, “Media and political polarization,”
Annual Re-view of Political Science , vol. 16, pp. 101–127, 2013.[21] D. Baldassarri and P. Bearman, “Dynamics of politicalpolarization,”
American sociological review , vol. 72, no. 5,pp. 784–811, 2007. [22] R. J. Dalton, “Social modernization and the end ofideology debate: Patterns of ideological polarization,”
Japanese Journal of Political Science , vol. 7, no. 1, pp. 1–22, 2006.[23] J. Levin,
Globalizing the community college: Strategiesfor change in the twenty-first century . Springer, 2001.[24] D. C. Hallin and P. Mancini, “Americanization, global-ization, and secularization,”
Comparing political commu-nication , pp. 25–44, 2004.[25] N. S. A. Gordon, “Globalization and cultural imperialismin jamaica: The homogenization of content and ameri-canization of jamaican tv through programme modeling,”
International journal of communication , vol. 3, p. 25,2009.[26] T. B. Ksiazek and J. G. Webster, “Cultural proximityand audience behavior: The role of language in pat-terns of polarization and multicultural fluency,”
Jour-nal of Broadcasting & Electronic Media , vol. 52, no. 3,pp. 485–503, 2008.[27] A. Flache and M. W. Macy, “Small worlds and culturalpolarization,”
The Journal of Mathematical Sociology ,vol. 35, no. 1-3, pp. 146–176, 2011.[28] R. Holton, “Globalization’s cultural consequences,”
TheAnnals of the American Academy of Political and SocialScience , vol. 570, no. 1, pp. 140–152, 2000.[29] C. Castellano, M. Marsili, and A. Vespignani, “Nonequi-librium phase transition in a model for social influence,”
Physical Review Letters , vol. 85, no. 16, p. 3536, 2000.[30] D. Vilone, A. Vespignani, and C. Castellano, “Orderingphase transition in the one-dimensional axelrod model,”
The European Physical Journal B-Condensed Matter andComplex Systems , vol. 30, no. 3, pp. 399–406, 2002.[31] F. V´azquez and S. Redner, “Non-monotonicity and diver-gent time scale in axelrod model dynamics,”
EPL (Eu-rophysics Letters) , vol. 78, no. 1, p. 18002, 2007.[32] E. Brigatti and A. Hern´andez, “Finite-size scaling anal-ysis of a nonequilibrium phase transition in the nam-ing game model,”
Physical Review E , vol. 94, no. 5,p. 052308, 2016.[33] N. Crokidakis and E. Brigatti, “Discontinuous phasetransition in an open-ended naming game,”
Journal ofStatistical Mechanics: Theory and Experiment , vol. 2015,no. 1, p. P01019, 2015.[34] R. Toral and C. J. Tessone, “Finite size effects inthe dynamics of opinion formation,” arXiv preprintphysics/0607252 , 2006.[35] C. J. Tessone, R. Toral, P. Amengual, H. S. Wio,and M. San Miguel, “Neighborhood models of minorityopinion spreading,”
The European Physical Journal B-Condensed Matter and Complex Systems , vol. 39, no. 4,pp. 535–544, 2004.[36] C. P. Herrero, “Ising model in scale-free networks: Amonte carlo simulation,”
Physical Review E , vol. 69,no. 6, p. 067109, 2004.[37] E. Brigatti and A. Hern´andez, “Exploring the onset ofcollective motion in self-organised trails of social organ-isms,”
Physica A: Statistical Mechanics and its Applica-tions , vol. 496, pp. 474–480, 2018.[38] M. A. Neto and E. Brigatti, “Discontinuous transitionscan survive to quenched disorder in a two-dimensionalnonequilibrium system,”