Reconciling long-term cultural diversity and short-term collective social behavior
Luca Valori, Francesco Picciolo, Agnes Allansdottir, Diego Garlaschelli
aa r X i v : . [ phy s i c s . s o c - ph ] A p r Reconciling long-term cultural diversity and short-term collective social behavior
Luca Valori , Francesco Picciolo , Agnes Allansdottir , Diego Garlaschelli Department of Chemistry, University of Siena,Via A. De Gasperi 2, 53100 Siena, Italy University of Siena, 53100 Siena, Italy Lorentz Institute for Theoretical Physics,University of Leiden, Niels Bohrweg 2,NL-2333 CA Leiden, The Netherlands (Dated: June 19, 2018)An outstanding open problem is whether collective social phenomena occurring over shorttimescales can systematically reduce cultural heterogeneity in the long run, and whether offlineand online human interactions contribute differently to the process. Theoretical models suggestthat short-term collective behavior and long-term cultural diversity are mutually excluding, sincethey require very different levels of social influence. The latter jointly depends on two factors: thetopology of the underlying social network and the overlap between individuals in multidimensionalcultural space. However, while the empirical properties of social networks are well understood, lit-tle is known about the large-scale organization of real societies in cultural space, so that randominput specifications are necessarily used in models. Here we use a large dataset to perform a high-dimensional analysis of the scientific beliefs of thousands of Europeans. We find that inter-opinioncorrelations determine a nontrivial ultrametric hierarchy of individuals in cultural space, a resultunaccessible to one-dimensional analyses and in striking contrast with random assumptions. Whenempirical data are used as inputs in models, we find that ultrametricity has strong and counterin-tuitive effects, especially in the extreme case of long-range online-like interactions bypassing socialties. On short time-scales, it strongly facilitates a symmetry-breaking phase transition triggeringcoordinated social behavior. On long time-scales, it severely suppresses cultural convergence byrestricting it within disjoint groups. We therefore find that, remarkably, the empirical distributionof individuals in cultural space appears to optimize the coexistence of short-term collective behav-ior and long-term cultural diversity, which can be realized simultaneously for the same moderatelevel of mutual influence. This also shows that long-range interactions may simultaneously enhancecoordination and sustain diversity.
PACS numbers: Valid PACS appear here
I. INTRODUCTION
How a society spontaneously organizes macroscopicallyfrom the microscopic, uncoordinated behavior of individ-uals is one of the most studied and exciting problems ofmodern science [1–3]. Collective social phenomena areobserved in different aspects of everyday life, includingthe onset of large-scale popularity and fashion (both of-fline [4, 5] and online [6, 7]), the existence of large fluc-tuations and herding behavior in financial markets [8–10], the spontaneous emergence of order in traffic andcrowd dynamics [11], the properties of voting dynam-ics [4, 12], the structure of country-wide communicationnetworks [3, 13, 14], the spreading of habits, fear, gos-sip, rumors, etc. [2, 11]. When collective phenomenaoccur, large parts of a population turn out to be globallycorrelated as a result of the combination of many local in-teractions, even if no centralized mechanism takes place.Importantly, the collective outcome is different from amere superposition of non-interacting individual behav-iors, and contrasts the ‘representative agent’ scenario of-ten postulated in economic theories. The different char-acteristics, choices and behaviors of individuals, ratherthan being ‘averaged out’ in the long run and at a largescale, may in some circumstances become amplified at the societal level [1]. The observation of these phenom-ena, also enabled recently by large-scale electronic plat-forms where people can exchange information with un-precedented speed and breadth (see Appendix for a moredetailed discussion), poses the question of whether thediversity of behaviors, attitudes and opinions is destinedto be progressively reduced in the long run. Naively, oneexpects that stronger collective social phenomena takingplace on short timescales may gradually result in morehomogeneous behaviors in the long term.This picture is reinforced by the fact that similar mech-anism are believed to be among the key driving forces ofboth collective social behavior [1, 7] and cultural con-vergence [15]. Various simplified models have been in-troduced to quantitatively simulate the fate of culturaldiversity and the dynamics of opinions in large groups[2, 11, 12]. In both cases, the main hypothesized mech-anisms are the tendency of social interactions to favorconvergence and consensus ( social influence [16]) and theinverse tendency of culturally similar individuals to in-teract more than dissimilar ones ( homophily ) [15]. Re-cently, the concept of homophily has been enriched withthe quantitative notion of bounded confidence , accordingto which people are not culturally influenced by too dis-similar peers [2, 17, 18]. Quantitatively, when the opin-ions or cultural traits of an individual are represented as ascalar or vector variable, the bounded confidence hypoth-esis results in two individuals being potentially influencedby each other only if the distance between their associ-ated variables is smaller than a certain threshold, rep-resenting the level of confidence or tolerance [2, 17, 18].This threshold, which in a simplified picture is assumedto have the same value ω across the entire population,quantifies how susceptible an individual is to possiblecultural influences. According to this picture, two indi-viduals can only influence each other if they are sociallytied and also sufficiently similar culturally: the effectivemedium of interaction is the overlap between the socialnetwork and the cultural graph connecting pairs of simi-lar individuals (see Appendix for an extended discussion).Rather than conveying a detailed picture of reality, mod-els of social dynamics aim at understanding the effectsthat different mechanisms proposed in social science mayhave when combined together and when taking place ata large-scale level. In particular, the importance of oneof the most popular models, proposed by Axelrod [15],resides in showing that social influence and homophily donot necessarily reinforce each other and determine a cul-turally homogeneous society. In fact, the model suggeststhat the persistence of cultural diversity in the long termis warranted by the inhibition of influence among dissim-ilar individuals, even if socially tied. However, if pluggedinto other models of social dynamics [2, 9, 11], the samemechanism prevents information diffusion across cultur-ally disconnected groups, and therefore also implies nocollective social behavior in the short term. We will givean explicit example of this effect simulating both short-and long-term dynamics on random data.According to the above results, the coexistence of long-term cultural diversity and short-term collective behav-ior is apparently a paradox, which can only be solvedby invoking different mechanisms at different timescales.However, here we show that, even without postulat-ing more complicated scenarios, the paradox can be ex-plained by taking into account an insofar ignored aspectof empirical multidimensional cultural profiles. II. THE HIERARCHICAL DISTRIBUTION OFINDIVIDUALS IN CULTURAL SPACE
Our study starts with the analysis of the large
Eu-robarometer dataset[19, 20], an official report of aquestionnaire-based survey[21, 22] of the European Com-mission, which allowed us to reconstruct the empiricalmultidimensional vectors of 13,000 individuals across 12European countries. In the Appendix we describe thedataset in more detail, and how the multiple-choice na-ture of the questionnaire allowed us to define, for each in-dividual i in country α , an F -dimesional vector ~v i whose k th component v ( k ) i represents the answer given by i toquestion k in the survey ( k = 1 , . . . F , where the num-ber of questions is F = 161). To obtain groups of equalsize, we sampled N = 500 individuals for each of the 12 countries, plus a thirteenth group of N = 500 individ-uals sampled all across Europe. We labeled each groupwith a Greek letter α = 1 , . . .
13. For each pair i, j ofindividuals we defined a normalized metric distance d ( k ) ij (0 ≤ d ( k ) ij ≤
1) between v ( k ) i and v ( k ) j , measuring how dif-ferent the answers given by i and j to question k were.We also defined an overall metric distance d ij (again suchthat 0 ≤ d ij ≤
1) between the entire sets of answers ~v i and ~v j given by i and j (see Appendix for all definitions).In addition to real data, we considered two types ofrandomized data which represent important null modelsproviding informative benchmarks throughout our anal-ysis. A first type of randomization (‘random answers’)simply consists in defining N random vectors, each ob-tained drawing F answers uniformly among the possiblealternatives. This simulates N individuals giving com-pletely random answers to the questionnaire, and doesnot depend on the empirically observed answers. Thisprovides a unique random benchmark against which allsampled groups can be compared, and corresponds to theusual initial specification in the Axelrod model [15] andsimilar models [2, 17, 18]. A second type of randomiza-tion (‘shuffled answers’) consists in randomly shuffling,for each of the F questions in the questionnaire, the realanswers given by the N individuals of the group consid-ered. In this case, different groups have different random-ized benchmarks, each characterized by its own probabil-ity distribution (determined by real data) of possible an-swers. This null model is very important, as it preservesthe number of times a particular answer was actuallygiven to each question (so it preserves ‘more fashioned’answers for each group), but destroys the correlations be-tween answers given by the same individual to differentquestions. [27]We analyzed several properties of real and randomizeddata, as a preliminary step before studying the impactof the empirical structure on short- and long-term dy-namics. A peculiar aspect of our multidimensional datais the possibility to investigate cross-correlations amongopinions, an information which is not available in one-dimensional studies. In particular, we studied whethersmall (large) differences between the answers given toquestion k imply small (large) differences between an-swers to question l by measuring the correlation ρ ( kl ) α be-tween d ( k ) ij and d ( l ) ij for all pairs k, l of questions and forall pairs i, j ∈ α of individuals belonging to group α (seeAppendix). We found that, whereas random and shuf-fled data display no significant correlation by construc-tion, real data are always characterized by a predomi-nance of strong positive correlations, plus a minority ofweak negative correlations. This pattern is analogous tothe ‘likes attract’ phenomenon: individuals with more be-liefs in common are more likely to agree on other opinions(strong positive correlation), while dissimilar individualstend to ignore, rather than repel, each other (weak neg-ative correlation). However, in this case we have an evi-dence of a deeper mechanism, since we know that individ- α Country µ α,real µ α,shuffled σ α,real σ α,shuffled µ α ) and standard deviation ( σ α ), for realand shuffled data, of inter-vector distances for the 12 groupssampled from European countries ( α = 1 ,
12; alphabeticalorder) plus the group sampled across Europe ( α = 13) andthe set of uniformly random data ( α = 14). uals in our data are socially unrelated. Therefore, ratherthan an effect of homophily and social influence, the ob-served result is the signature of intrinsic inter-opinioncorrelations in a single individual.The observed inter-opinion correlations have impor-tant effects on the distribution of individuals, i.e. of thevectors { ~v i } , in cultural space. We find (see Table I)that the average inter-individual distance µ α ≡ h d ij i ij ∈ α of random data is larger than real data, while it is easy toshow theoretically (and confirm by looking at the mea-sured values) that real and shuffled data always have thesame value of µ α , i.e. µ α,real = µ α,shuffled . This meansthat the observed ‘attraction’ among opinions does notimply, as one would naively expect, that the empiricalvectors { ~v i } are closer to each other in cultural spacethan shuffled data. However, real and shuffled data dif-fer significantly in other properties of the distributionof vectors in cultural space. A first difference is thatreal distances are much more broadly distributed thanshuffled ones. This can be inspected by measuring theintra-group variance σ α ≡ h d ij i ij ∈ α − h d ij i ij ∈ α . As canbe seen from Table I, σ α,real is roughly twice as largeas, and more variable than, σ α,shuffled (see Appendixfor more details). Further important higher-order differ-ences between real and randomized data can be charac-terized by performing a hierarchical clustering algorithmof the vectors { ~v i } , which represents the latter as leaves ofa dendrogram where culturally closer individuals have alower common branching point. This is shown in fig.1 forreal, shuffled, and random data. As one can clearly see,the dendrogram for real data is well structured in sub-branches nested within branches, indicating that culturalspace is heterogeneously populated by dense communi-ties of similar individuals, separated by sparsely occupiedregions. The hierarchical character of this distributionshows that denser regions are iteratively fragmented into a L Real b L Shuffled c L Random
FIG. 1: Dendrograms resulting from the application of anaverage linkage clustering algorithm to the cultural vectors { ~v i } , represented as leaves of the tree along the horizontalaxis. a) Real Germany data. b) Shuffled Germany data. c) Random data. denser regions nested within them. This peculiar organi-zation indicates that the original distances are (nearly) ultrametric [23], i.e. the tree-like representation renderedby the dendrogram is not just an artifact of the clus-tering algorithm, but a natural property of the data.This means that the height of the first branching pointconnecting two individuals i and j approximately corre-sponds to the original distance d ij between ~v i and ~v j . Bycontrast, the dendrograms for shuffled and real data aretrivially structured, with no well-defined internal separa-tion between different hierarchical levels. In this case thedendrogram is not representative of the original distribu-tion of vectors, and is merely an uninformative outcomeof the algorithm which is forcing non-ultrametric datainto a tree-like description. In such a situation, the verti-cal dimension of the dendrogram loses it correspondencewith the original inter-vector distances, and provides ahighly distorted image of the latter. Thus, we find thatthe broader distribution of real distances (with respectto shuffled ones) is implied by the ultrametric structure,characterized on one hand by an increased frequency ofboth nearby vectors (representing individuals within thesame branch of the dendrogram), and on the other handby an increased frequency of distant ones (representingindividuals belonging to different branches). By contrast,shuffled data generate vectors with the same average dis-tance but more uniformly and non-ultrametrically dis-tributed in cultural space. III. LOCAL AND GLOBAL LEVELS OFINFLUENCE
The ultrametric hierarchy discussed above has impor-tant static and dynamic consequences. As we show in theAppendix, the branches of the dendrogram ‘cut’ horizon-tally at a distance ω coincide with the connected compo-nents of the ω -dependent cultural graph we defined in thebeginning. It is interesting to study, as a function of ω ,the density of links f α ( ω ) (which is nothing but the CDFof the distance distribution) and the size fraction s α ( ω )of the largest connected component in the cultural graph,which represent a local and a global measure of influenceamong the individuals of group α respectively. The re-sulting curves are shown in fig.2a-b. For both quantities,we observe large differences between real and random-ized data. In particular we find that, for a given valueof α , real data are characterized by higher levels of localand global influence than shuffled and random data. Inorder to understand whether the differences among thecurves in figs.2a and 2b can be simply traced back tooverall differences in the average values ( µ α ) and vari-ances ( σ α ) of the inter-vector distances, in fig.2c-d weshow f α and s α when plotted as a function of the stan-dardized parameter z ≡ ( ω − µ α ) /σ α . We find (see fig.2c)that all the f α ( z ) plots collapse onto a single univer-sal curve, which is indistinguishable from the cumulativedensity function (CDF) of the standard Gaussian distri-bution f ( z ) ≡ R z −∞ dx ( e − x / ) / √ π . In other words, ineach group α the distances are normally distributed, withgroup-specific mean µ α and variance σ α . This means thatall the empirical differences in link density among groupsare taken care of after rescaling the distance, so that thecontrol parameter z completely specifies the density ofrealized cultural channels of any group. Thus, if culturalchannels were placed uniformly among individuals as ina homogeneous random graph, one would also observean analogous universal collapse of the s α ( z ) curves andof any other topological property. By contrast, as canbe seen in fig.2d, this approximately occurs only in theshuffled case, but not for real data. This result indicatesagain a nontrivial distribution of real cultural vectors,and singles out differences across the sampled groups thatare not simply explained in terms of an overall variability.In particular, the universality of the link density functionobserved in fig.2c does not imply a universal structure ofconnected components, due to correlations between pairsof edges generated by the correlations between distances.In other words, even after standardizing the local level ofmutual influence, real data continue to differ significantlyin their global level of influence. Therefore any processwhich depends on the cultural distance between individ-uals might have very different global outcomes even whentaking place on locally identical structures.All the above results show that even randomly sam-pled individuals (as the ones in our database) are notcharacterized by uniformly random cultural vectors. [28]While this is not surprising, the peculiar hierarchical dis- ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ‰ Real + Shuffled ƒ Random a0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.0 f H Ω L ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰+ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + +ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ‰ Real + Shuffled ƒ RandomNormal c - - - f H z L ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ƒ ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ‰ Real + Shuffled ƒ Random b0.1 0.2 0.3 0.40.00.20.40.60.81.0 Ω s H Ω L ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰+ + + + + + ++ + + + + + ++ + + + + + ++ + + + + + ++ + + + + + ++ + + + + + + ++ + + + + + ++ + + + + ++ + + + + + +ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ‰ Real + Shuffled ƒ Random d - - - - - s H z L FIG. 2: Local and global measures of influence in culturalgraphs obtained from real and randomized data. a) Fractionof direct influential interactions (link density) f α as a func-tion of the threshold ω . b) Fraction of the largest connectedcomponent s α emerging from indirect influential interactionsas a function of ω . c) Link density f α as a function of therescaled threshold z ≡ ( ω − µ α ) /σ α . The black solid line is thecumulative density function (CDF) of the standard Gaussiandistribution. d) Fraction of the largest connected component s α as a function of the rescaled threshold z . tribution implied by empirical inter-opinion correlationsis highly nontrivial, and unpredictable a priori . Since,as we show in what follows, the dynamics of opinionsand culture is strongly dependent on initial conditions,it is important to investigate how the predictions of pop-ular models change when the empirically observed dataare considered as the starting configuration, rather thanthe ordinarily postulated [2, 15, 17, 18] random (or evenonly uncorrelated) cultural vectors. To this end, in whatfollows we study how the empirically observed ultramet-ric structure affects the predictions of models simulatingboth short-term and long-term dynamics, with a partic-ular interest in exploring the effects on the coexistenceof cultural heterogeneity and collective social phenom-ena. Rather than considering one or more (unavoidablyarbitrary) specifications of possible social networks thatideally start connecting the (initially non-interacting) in-dividuals of a group, we choose to establish a unique up-per bound for the achievable level of influence, where thesocial network is virtually replaced by a complete graph.This choice corresponds to selecting the maximum levelof influence on the social side of the problem, and let-ting the value of the cultural threshold ω uniquely deter-mine whether two individuals can influence each other( d ij < ω ) or not ( d ij > ω ) according to the bounded con-fidence hypothesis. As discussed in the Appendix, thenew possibilities of interactions that have been recentlybecome available on online platforms, where individualsinfluence each other bypassing social ties, are modifyingthe traditional scenario and leading us closer to this ex-treme ‘complete graph’ setting. In any case, rather thanthe dynamical outcome in absolute terms (which stronglydepends on the specification of the underlying network),our main interest is the comparison, on the same network,of the outcome implied by real cultural vectors with thatimplied by randomized data. IV. SHORT-TERM COLLECTIVE SOCIALBEHAVIOR
We first study the effects of the empirical structureof real opinions on short-term collective social behavior.We consider a simple prototypic model where, on shorttimescales, cultural vectors do not evolve but nonethelessdetermine the choices that individuals make under theinfluence of each other. To this end, we extend the Cont-Bouchaud (CB) model [9], originally proposed to modelherding effects in financial markets, to a more general‘coordination model’ which incorporates a dependenceon real cultural vectors { ~v i } (see Appendix). For eachgroup in our analysis, we consider a situation where in-dividuals are asked, for instance in democratic elections,public referenda, financial markets, online surveys, etc.,to make a binary choice such as yes/no, buy/sell, ap-prove/reject, left/right etc. We can represent the choiceexpressed by the i -th individual as φ i = ±
1. The effectsof mutual influence and bounded confidence are modeledby allowing pairs of individuals whose cultural distance d ij is smaller than a threshold ω (which is the only pa-rameter of the model) to exchange information beforemaking their choices. As a result of this information ex-change, we assume that all the agents belonging to thesame connected component of the resulting ω -dependentcultural graph (see Appendix) collectively agree on thechoice to make. If A labels a connected component of thegraph, the choice of all agents belonging to A is the same( φ i = φ A ∀ i ∈ A ), while different connected componentsmake statistically independent choices.The overall outcome of the process (e.g. the result ofthe survey/referendum/election) is the sum of individualpreferences, and can be quantified by the average choiceΦ = 1 N N X i =1 φ i = 1 N X A S A φ A (1)where the second sum runs over all connected compo-nents, and S A is the size of component A . The sign ofΦ reflects the choice of the majority, and a key propertycharacterizing the outcome of the model is the proba-bility P ω (Φ) that the average choice takes the particu-lar value Φ, for a given value of ω . Following the pro-cedure described in Appendix, we computed P ω (Φ) forvarious values of ω (from ω = 0 to ω = 1 in incrementsof 0 .
01) and for all the 13 groups in our dataset (both realand shuffled), plus the completely random set. In fig.3awe report the results for real Germany data. As can beseen, there exists a critical value ω c (in the case shown, ω c = 0 . ± .
01) such that, for ω < ω c , P ω (Φ) is symmet-ric about zero (as for ω = 0) and, for ω > ω c , P ω (Φ) hastwo symmetric peaks. Right at ω = ω c , P ω (Φ) displaysa flattened region. This behavior is typical of symmetry-breaking phase transitions. Here the order parameter of ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ a + Ω = ƒ Ω c = ‰ Ω = - - F P Ω H F L ƒ ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ƒƒƒƒƒ ƒƒ ƒƒƒƒƒƒ ƒƒƒƒƒƒƒ ƒƒƒƒƒƒ ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ +++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++ + + ++ ++ ++++++++++++++++++++‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰‰ ‰‰ ‰‰ ‰‰ ‰‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ Real + Shuffled ƒ Random b - - Ω F ± H Ω L FIG. 3: At a critical confidence level, a spontaneous break-ing of choice symmetry occurs. a) When our ‘coordinationmodel’ is simulated on real data (in the example shown, theGermany group), we observe an abrupt change in the proba-bility P ω (Φ) of a collective consensus at a critical confidencevalue ω c . For ω < ω c , individual choices are uncorrelated andsum up to a vanishing global outcome Φ = 0, at which P ω (Φ)has a single peak. For ω > ω c , local interactions result inglobal correlations that spread across the entire system, anda macroscopically coordinated output, whose probability ispeaked about the two nonzero values Φ ± ( ω ), emerges. Rightat ω = ω c , P ω (Φ) displays a flat region typical of criticalphenomena. b) The most probable value of Φ is the orderparameter of the phase transition. For ω < ω c it is vanishing,while for ω > ω c it branches into the two symmetric valuesΦ ± ( ω ). In addition to the results for real Germany data, herewe also show the results for shuffled and random data. Realdata always have a lower critical threshold than randomizeddata, indicating an enhanced possibility to behave collectively.All the other groups show the same behavior. the transition is the most probable value(s) Φ ± of Φ: for ω < ω c one has Φ + ( ω ) = Φ − ( ω ) = 0, while for ω > ω c one has two symmetric values Φ − ( ω ) < < Φ + ( ω ) withΦ − ( ω ) = − Φ + ( ω ). This is shown in fig.3b, where we alsoplot the behavior for shuffled and random data. [29]This analysis allows to measure the critical thresholds ω c for all the groups we considered (see fig.4). Note thatsmaller (larger) values of ω c require smaller (larger) levelsof influence between individuals in order to trigger collec-tive behavior. Therefore ω c represents a novel measureof the resistance of a social group to act collectively. Im- RandomShuffledReal P o r t ug a l G r eece Ir e l a nd S p a i n E u r op e I t a l y G e r m a ny UK L ux e m bou r g B e l g i u m F r a n ce N e t h e r l a nd s D e n m a r k - Group C r iti ca lt h r e s ho l d H Ω c L FIG. 4: Critical thresholds representing resistance to collec-tive behavior of the sampled groups, for real, shuffled andrandom data. For real data, the values of ω c are not consis-tent with each other within the error bars, indicating differentresistances to coordination across the sampled groups. Theyare however always smaller than randomized data, indicatingthat the empirical ultrametric distribution in cultural spacesystematically facilitates the onset of collective behavior. portantly, we found that the thresholds for shuffled dataare always larger than those for real data (see fig.4), andthe ones for random data are even larger. This showsthat empirical inter-opinion correlations, which are re-sponsible for the ultrametric distribution of individuals incultural space, strongly facilitate collective social behav-ior by systematically lowering the resistance to coordina-tion. While for shuffled and random data all thresholdsare equal within errors, the entity of the enhancement ofcollective behavior in real data varies significantly acrossthe sampled groups and determines non-universal valuesof the thresholds. In general, ω c also represents a fun-damental threshold for any dynamical process dependenton cultural data. Any mechanism taking place within acultural distance smaller than ω c will not propagate tothe whole network, while if the interaction range is largerthan ω c the information can percolate the entire system.This simple model indicates that, depending on thelocal interaction range, individual differences can eitherbe ‘averaged out’ and disappear at the macroscopic levelor give rise to a collectively coordinated behaviour. Un-derstanding this transition in real societies is one of thefundamental open questions of modern social science [1].In economics, this problem is related to whether it is le-gitimate to use the concept of ‘representative agent’ asan idealized individual that makes the average choice ofthe society. Contributions from different scientific com-munities to this ongoing debate gave very different pointsof view about the subject. Our simplified model, whensimulated on real data, suggests that both regimes arepossible, and that a simple local parameter can triggervery different global outcomes. In particular, the vari-ance of the collective outcome, which grows with theseparation between the peaks of P ω (Φ), can either de-cay to zero or be amplified macroscopically. In the latter case, the final outcome is collective (or, in the case ofvotes, democratic) in a ‘strong’ sense: a large portion ofthe population makes a coordinated action, and the ma-jority’s choice reflects a truly collective consensus. Bycontrast, in the former case the result is collective ordemocratic in a ‘weak’ sense: there are several groups ofpeople making different choices, and the majority’s ac-tion does not reflect a collective agreement, but rather astatistical fluctuation about a 50%-50% tie [12]. Theseconsiderations indicate that a good measure of the levelof collective behavior achievable for a given value of ω is the width of P ω (Φ). Thus we can define the standarddeviation C ( ω ) ≡ σ ω (Φ) = vuutX A (cid:18) S A N (cid:19) ω (2)as a measure of short-term social coordination . The lat-ter equality in the above formula (see Appendix for arigorous proof) states that, intriguingly, C ( ω ) is uniquelydetermined by the sizes { S A } of the connected compo-nents of the underlying cultural graph obtained for thatparticular value of ω , and is therefore actually indepen-dent of the dynamical model considered. This quantitywill be useful in what follows. Note that C ( ω ) rangesbetween 0 and 1. If ω = 0 (no coordination), its value is C ( ω ) ≃ / √ N (as follows from the Central Limit Theo-rem) and vanishes for large N . At the opposite extreme,if ω = 1 (complete coordination) then C ( ω ) = 1 which isas large as the standard deviation of the individual choice φ i , and remains finite when N → ∞ . V. LONG-TERM CULTURAL DIVERSITY
We now take an evolutionary perspective and focus ona longer temporal scale over which the cultural vectorsthemselves can change. In this case we use a modifiedversion [17] of the popular Axelrod model [15], which isdesigned to simulate the evolution of vectors of culturaltraits on social networks, again in a way that real datacan enter into the model.[30] A detailed discussion of themodel can be found in the Appendix. In an elementarytime-step, two individuals i and j belonging to the samesampled group are randomly selected. If the generalizedoverlap o ij ≡ − d ij (where d ij is the above-defined dis-tance between the vectors ~v i and ~v j ) is smaller than orequal to θ , no interaction takes place. Otherwise, withprobability equal to o ij , the two individuals interact: acomponent v ( k ) j , chosen randomly among the componentswhere ~v i and ~v j differ, is changed and set equal to i ’s cor-responding component: v ( k ) j = v ( k ) i . Otherwise nothinghappens, and two other individuals are selected. Theserules implement the two basic mechanisms of social influ-ence (interacting actors tend to converge culturally) andhomophily (similar individuals interact more frequently).Note that, in line with our previous analysis, we are as-suming that every pair of individuals can interact, i.e.the underlying social network is a complete graph. Inthe allowed final configurations, any two cultural vectorsare either completely identical or separated by a distancelarger than ω ≡ − θ , and the average h N D i ω (over manyrealizations) of the number N D of distinct vectors in thefinal stage, or equivalently the fraction D ( ω ) ≡ h N D i ω N (3)is a convenient way to measure the long-term cultural diversity as a function of ω .We ran several realizations of the model by taking bothreal and randomized cultural vectors { ~v i } as the startingconfiguration. As we show in the Appendix, we find thatreal data are those that achieve the largest level of long-term cultural heterogeneity (value of h N D i ). Indeed, forreal data the realized value of h N D i is the largest possi-ble ( h N D i ≈ N C ) indicating that cultural convergence isconfined within the initial connected components, each ofwhich eventually becomes a single cultural domain. Bycontrast, in randomized data there are less final culturaldomains than initial connected components, indicatingthat the latter often ‘merge’ into larger cultural domains.The reason for the remarkably different behavior of realand randomized data is, once again, the ultrametric char-acter of the former. As we show in fig.5, ultrametricityimplies that the branches obtained cutting the real-datadendrogram at some value of ω will collapse into a sin-gle cultural vector. This means that the initial structureof the dendrogram above ω will be ‘frozen’ and unaf-fected by cultural evolution. This confines cultural con-vergence locally within the lower branches. By contrast,in randomized data the lack of ultrametricity implies thatbranches are not well separated, so that the local conver-gence of vectors within a branch reduce the separationof the branch itself from nearby branches. Thus in thiscase branches are unstable, and often merge modifyingthe entire structure globally.We can combine the above findings with our previousresults about collective behavior. In particular, given agroup of individuals, we can measure both the short-termsocial coordination C ( ω ) defined in eq.(2) and the long-term cultural diversity D ( ω ) defined in eq.(3) for variousvalues of ω . Then we can plot D ( ω ) versus the value C ( ω ) obtained for the same ω , as in fig.6. If we look atrandom data, we retrieve the naive result that the co-existence of cultural heterogeneity and social collectivebehavior is impossible, since we have either D ≈ C ≈
0. Note that a cultural graph defined among randomvectors is approximately equivalent to a random graph,whose density is completely determined by ω through therelation shown in fig.2a. Therefore the results we show forrandom vectors coincide with the standard results thatwould be obtained by simulating the Cont-Bouchaud andAxelrod models on random graphs, for various densityvalues. By contrast, we find that real cultural vectors al-low high simultaneous levels of short-term coordinationand long-term diversity, including the approximately bal- a L Initial H real L b L Final H simulated L c L Initial H real L d L Final H simulated L FIG. 5: The hierarchical structure implied by inter-opinioncorrelations constrains cultural evolution. a) The real, hier-archically organized cultural vectors for the Germany group(the same as shown in fig.1a) are considered as the initialstate of the modified Axelrod model, and a confidence level(corresponding to the horizontal line below which the shadedregion originates) is imposed. b) Due to ultrametricity, inthe corresponding final state of the model all the individualswithin a common shaded branch in the initial dendrogramcollapse to the same cultural vector, with negligible effects onthe upper part of the dendrogram. c) The same initial stateas above is considered, but a lower confidence level is imposed. d) Correspondingly, the final state of the model consists ofa larger number of distinct cultural vectors, each containingon average less individuals with collapsed vectors. Thus thenumber of distinct final vectors (the leaves of the final den-drogram) coincides with the number of branches intersectingthe horizontal line in the initial dendrogram. If shuffled orrandom opinions are taken as the initial state of the model(not shown), this is no longer true since the convergence ofcultural vectors also affects the dendrogram’s structure abovethe horizontal line, signalling a lack of ultrametricity. anced regime C ≈ D ≈ / ω ≈ .
17. Shuf-fled data follow an intermediate curve, showing that theheterogeneous frequencies of real opinions and the cor-relations among the latter both play a significant rolein enhancing the coexistence of diversity and coordina-tion. Thus, surprisingly, we find that empirical hierarchi-cal correlations simultaneously enhance collective behav-ior and sustain cultural heterogeneity. While the incom-patibility of these two phenomena holds for randomizeddata, which represent the usual specification of dynami-cal models, it is violated by real data. This remarkableresult highlights the scarce predictive power of modelsthat consider random specifications, and shows the im-portance of empirical analyses of high-dimensional cul-tural vectors, which offer the unprecedented possibilityto explore cross-correlations among opinions and theirconsequences. ƒƒƒƒƒƒƒƒƒƒ ƒ ƒ ƒ+++++++ + + + + + + ++++++++‰‰‰‰‰‰‰‰‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰‰‰‰‰‰‰‰‰ ‰ Real + Shuffled ƒ Random - term social coordination H C L L ong - t e r m c u lt u r a l d i v e r s it y H D L FIG. 6: Phase diagram summarizing our results. The long-term cultural diversity D is shown as a function of short-termsocial coordination C for real, shuffled and random data. Ifrandom cultural vectors are considered, cultural heterogeneityand collective behavior are mutually excluding: one has either D ≈ C ≈
0. This approximately corresponds to the tra-ditional situation explored when considering a random graphof interaction among individuals. By contrast, real culturalvectors allow high simultaneous levels of short-term coordi-nation and long-term diversity, including the approximatelybalanced regime C ≈ D ≈ /
2. Shuffled data follow an inter-mediate curve, showing that the heterogeneous frequencies ofreal opinions and the correlations among the latter both playa significant role in enhancing the coexistence of diversity andcoordination in the real world.
VI. CONCLUSIONS
By using a large detailed dataset, we have character-ized the empirical properties of the large-scale distribu-tion of individuals in multidimensional cultural space.We found that real inter-opinion correlations organizeindividuals hierarchically and ultrametrically in culturalspace, a result which is not retrieved when randomized orone-dimensional opinions are considered. These proper-ties strongly determine the exploitable network of inter-actions that is expected to arise as a result of the boundedconfidence hypothesis, according to which individuals areonly influenced by culturally similar peers. Ultrametric-ity has profound and nontrivial consequences on short-and long-term cultural dynamics. In the short term, ifone assumes that consensus can be reached by individualswith sufficiently similar opinions, we found the existenceof a symmetry-breaking phase transition where collec-tive behavior arises out of purely local interactions. Thecritical threshold of this transition is remarkably lowerin real data than in randomized cases, indicating thatultrametricity enhances short-term collective behavior.However, in the long term the same ultrametric propertysuppresses cultural convergence by restricting it withingdisjoint domains, implying a strong sensitivity to initialconditions. These opposite effects imply that, whereasin random data the coexistence of short-term coordina- tion and long-term diversity is unfeasible, in real datait is strongly enhanced and can be achieved in a broadregion of parameter space. Thus the apparent paradoxof the coexistence of short-term collective social behaviorand long-term cultural diversity might have, as a simpleand parsimonious explanation, the empirically observedhierarchical distribution of individuals in cultural space.
Appendix A: Social networks and cultural graphs:the online shift
Here we discuss how social influence, homophily andbounded confidence are believed to affect social dynam-ics, and how offline and online environments are expectedto contribute differently to the process. As we men-tioned in the main text, the bounded confidence hypoth-esis states that two individuals are potentially influencedby each other only if the distance between their culturalvariables (or vectors) is smaller than a certain threshold ω . Thus, while social influence takes place on a socialnetwork connecting individuals, bounded confidence in-volves a different graph, that we denote as the culturalgraph , where pairs of individuals separated by a distancesmaller than the confidence ω are connected by ‘culturalchannels’, irrespective of whether they are neighbors ina social network. When combined together, these hy-potheses imply that the actual network of interactions isgiven by the overlap between social ties and (confidence-dependent) cultural channels. If s ij and c ij ( ω ) denote theelements of the adjacency matrix of the social and cul-tural network respectively (equal to 1 if a link betweenvertices i and j is there, and 0 otherwise), the overlapnetwork is described by an adjacency matrix with en-tries a ij ( ω ) = s ij c ij ( ω ), where c ij ( ω ) = 1 if d ij < ω and c ij ( ω ) = 0 if d ij > ω .Models that simulate the evolution of societies musttherefore be complemented by empirical analyses charac-terizing both social networks and cultural graphs. How-ever, while a huge literature is devoted to the study ofreal networks formed by social ties (using both tradi-tional small-scale surveys [24] and more recently large-scale communication data [13, 25]), little is known aboutthe empirical properties of those formed by cultural chan-nels. As a consequence, when considering models of opin-ion dynamics, social networks have been so far consideredas proxies for the actual interaction graphs, i.e. a ij ≈ s ij ,which amounts to assume c ij ≈ ω = + ∞ ) for all pairs of individu-als. This assumption is helpful in the traditional offlinesituation where social ties are expected to be dominantover cultural channels, for instance when social groupsare formed independently of the cultural traits of people(e.g. acquaintances made in public schools). However,it prevents our understanding of the opposite extreme,i.e. when people look for culturally similar peers, thatthey do not know initially, to interact with (for instanceby joining an open discussion group on a focused topicof interest). In such a situation the empirical knowledgeof c ij ( ω ) may become even more important than that of s ij . While these non-social interactions used to representonly a secondary means of opinion diffusion up to someyears ago, they are now becoming more and more per-vasive as novel electronic platforms are being developedand used at a large scale [3, 6, 7, 14]. Indeed, in recentyears many new possibilities of interaction have rapidlyemerged, such as the exchange of opinions through theWWW (on-line forums, blogs, discussion groups, etc.)and other media. In these platforms, people already in-terested in a topic search new peers (who they may nevermeet physically afterwards) to discuss with, experiencingnovel ways by which opinions can interact. These interac-tions can be even stronger than those occurring throughdirect knowledge, as virtual communities gather togetherpeople with oriented and focused interests, who often rec-ognize each other as the most natural and qualified peersto share ideas with, even if no direct knowledge existsbetween them.The above picture suggests an extension of the idea ofhomophily, according to which ‘likes attract’, to an onlinesetting. When these additional possibilities of interactionare considered, cultural channels may become the domi-nant means of interaction, as people with similar interestsare more likely to access the same platform(s) and ex-change opinions more frequently than culturally dissimi-lar individuals. In this opposite extreme, the ‘social’ net-work through which opinions can in principle interact isvirtually replaced by a complete graph (i.e., s ij = 1 ∀ i, j )where everybody is connected to everyone else (in socialspace, the interaction becomes infinite-ranged). Culturalgraphs therefore become a natural proxy for the actualinteraction graph: a ij ( ω ) ≈ c ij ( ω ). Despite its increas-ing importance, our understanding of this novel type oflong-range opinion dynamics, which bypasses social tiesand is dominated by the structure of real opinions andcultural graphs, is incomplete. Our analysis bridges thisgap by using real data about the opinions, beliefs andattitudes of thousands of Europeans to produce culturalgraphs where we can investigate the outcome of infinite-range, online-like dynamics on both short and long timescales. Appendix B: Definition of cultural vectors anddistances
Here we describe how we reconstructed real multidi-mensional opinions from empirical data. We note that,for our analysis to be insightful, we need to access theopinions of a set of real individuals who do not know eachother and are well separated socially. This is essential inorder to separate the purely cultural dynamical effectsfrom ordinary social influence effects (producing conver-gent cultural traits) that might be already present in thedata if the individuals are sampled nearby in a social net-work. For this reason, we focused on a large dataset that is specifically designed to survey a number of beliefs andopinions across Europe, and based on standard samplingprotocols ensuring that individuals are selected avoidingthe bias due to (among other factors) social closeness.This dataset [21, 22], an official release of the Eurobarom-eter project [19, 20], reports (in its 1992 snapshot[31]) theresults of face-to-face interviews where about 13,000 in-dividuals across 12 European countries[32]) were askedto fill a questionnaire containing several multiple-choicequestions. These questions were designed to capture arange of individual beliefs and attitudes towards vari-ous scientific topics, thus surveying the ‘Public Under-standing of Science’ across Europe. Besides probing thegeneral level of scientific awareness in the individuals,the questionnaire focused on issues that were considered‘hot topics’ in relation to European integration, e.g. theintroduction of novel biotechnologies, the role of scien-tists in the dissemination of their research results, variousbioethical questions, etc. The database is invaluable inorder to study the real multidimensional organization ofopinions, as well as its dynamical consequences. In par-ticular, it allows us to establish a previously unavailableempirical reference for theoretical models, such as theaforementioned one proposed by Axelrod, where individ-uals are represented as vectors of cultural traits evolvingthrough the interaction with peers.Raw data, originally arranged in a SPSS spreadsheetfile of N = 13 ,
000 rows where row i reported the (numer-ically coded) answers of the i -th individual to F = 161multiple-choice questions, were transformed into N F -dimensional vectors { ~v i } . Each component v ( k ) i of thesevectors was given a value such that the correspondingcontribution d ( k ) ij to the overall distance d ij = 1 F F X k =1 d ( k ) ij (B1)is in the range [0 , d ( k ) ij = 0 representing i and j giving an identical answer to question k and d ( k ) ij = 1representing i and j giving opposite answers. For ‘met-ric’ questions, where answers were possible in an equallyspaced scale of Q k possibilities, the maximum informa-tion is retained by mapping the original answers to thepossible values v ( k ) i = 0 , Q k − , Q k − , . . . , d ( k ) ij ≡ | v ( k ) i − v ( k ) j | . For non-metric ques-tions (associated to Q k unordered possible alternatives),‘opposite answers’ simply means ‘different answers’, andwe therefore mapped the possible values of v ( k ) i to Q k arbitrary symbols and defined d ( k ) ij ≡ v ( k ) i = v ( k ) j and d ( k ) ij ≡ d ij = 1 − o ij /F where o ij isthe overlap (number of components with identical value)0between ~v i and ~v j , a commonly used notion of culturalsimilarity [15] as mentioned in the main text. Note thatfor binary answers (such as ‘yes’/‘no’) the metric andnon-metric definitions coincide.Each individual belongs to one of the 12 Europeancountries in the dataset. This information allowed us togenerate groups of individuals sampled either from thesame country, or from different ones. We labeled dif-ferent groups of individuals with different Greek letters( α, β, . . . ), and use the notation i ∈ α to indicate an indi-vidual i belonging to group α . In order to deal with sam-ples of equal size, we selected N = 500 individuals percountry[33] and generated 12 groups accordingly. Thisreproduces a situation where, for instance, people withthe same language join a medium-sized online discussiongroup mediating electronic, non-social interactions men-tioned above. Similarly, it mimics individuals that arephysically ‘put together’ to participate to a discussiongroup or to a social experiment. Finally, this choice al-lows to establish an upper bound for the cultural ho-mogeneity predicted by the ‘traditional’ dynamics tak-ing place on any possible social network connecting thesame individuals. We also generated an additional thir-teenth group with N = 500 individuals sampled from allthe 12 European countries, and denoted it as the Europe group. This reproduces a situation analogous to the onedescribed above, but where several individuals across Eu-rope can form a group irrespective of their nationalities,e.g. using a common language such as English.
Appendix C: Measuring inter-opinion correlations
Our multidimensional data allowed us to investigatecross-correlations among opinions, an information whichis not available in one-dimensional studies. We measuredthe covariance matrix between d ( k ) ij /F and d ( l ) ij /F , whoseentries read σ ( k,l ) α ≡ h d ( k ) ij d ( l ) ij i ij ∈ α − h d ( k ) ij i ij ∈ α h d ( l ) ij i ij ∈ α F (C1)where h·i ij ∈ α denotes an average over all pairs of individ-uals in group α . From the above matrix it is possible toobtain the inter-opinion correlation matrix, whose entriesread ρ ( k,l ) α ≡ σ ( k,l ) α σ ( k,k ) α σ ( l,l ) α (C2)and range between − k, l ) characterized by strong positive cor-relations ( ρ ( k,l ) α ≈ a L Inter - opinion correlation matrix H Germany, real L - b L Inter - opinion correlation matrix H Germany, shuffled L - FIG. 7: Colour plots of Inter-opinion correlation matrices ρ ( k,l ) α for the Germany group. a) real data. b) shuffled data. negative correlations ( ρ ( k,l ) α . i and j giving similar/different answers to ques-tion k (small/large d ( k ) ij ) tend to give similar/differentanswers to question l as well (small/large d ( l ) ij ), while theopposite outcome (small d ( k ) ij and large d ( l ) ij ) occurs muchless frequently. As we show in fig.7b, the inter-opinioncorrelation matrix for shuffled data lacks any structure,indicating the absence of statistically significant correla-tions (obviously, the same is true for random data, notshown).The elements of the covariance matrix determine the1intra-group variance σ α ≡ h d ij i ij ∈ α − h d ij i ij ∈ α = X k,l σ ( k,l ) α (C3)Note that for shuffled data we have σ α,shuffled = X k σ ( k,k ) α,shuffled = X k σ ( k,k ) α,real (C4)since σ ( k,k ) α,shuffled = σ ( k,k ) α,real and σ ( k,l ) α,shuffled = 0 for k = l ,while for real data we have σ α,real = σ α,shuffled + X k = l σ ( k,l ) α,real > σ α,shuffled (C5)where the last inequality comes from the observed posi-tivity of P k = l σ ( k,l ) α,real . Therefore, even if the distributionof inter-vector distances has the same average value inreal and shuffled data, real distances are more broadlydistributed than shuffled ones. Appendix D: Link density and largest connectedcomponents
The ultrametricity of real inter-vector distances impliesthat if we ‘cut’ the dendrogram of cultural vectors atsome height ω we obtain a set of disconnected branches,within which individuals are separated by a cultural dis-tance smaller than ω , and across which individuals areseparated by a distance larger than ω . In other words, weobtain the connected components of the cultural graphdefined by linking pairs of individuals separated by a dis-tance lower than a certain threshold ω . The conceptof bounded confidence implies that individuals belongingto different connected components of the cultural graph,even if linked by a social tie, cannot interact. Thereforein a fragmented cultural graph information (intended asmutual influence) can only diffuse locally. A necessarycondition in order to have a global spread of informationis that cultural channels form a giant connected compo-nent spanning (a finite fraction of) the N individuals ina given group. The fraction s α ( ω ) of vertices spannedby the largest connected component for a given value of ω represents an upper bound for the fraction of individ-uals in group α that can mutually influence each other,through either direct or indirect interactions, if the con-fidence threshold is set to ω . Therefore s α ( ω ) is a globalmeasure of potential influence, capturing how local inter-actions combine together at a large-scale level. It is alsoimportant to consider a purely local measure of influence,i.e. the average probability that any two individuals caninfluence each other through a direct interaction. To thisend, we also measure the density f α ( ω ) of realized cul-tural channels, i.e. the fraction of pairs of individuals ingroup α closer than ω . Note that f α ( ω ) is simply thecumulative density function (CDF) of the distance dis-tribution, i.e. it counts how many pairs of individuals are at distance smaller than ω . For ultrametric data, italso coincides with the fraction of pairs of cultural vectorswithin the same connected branches, when the dendro-gram is cut at the height ω as discussed above. By con-trast, for randomized data both f α ( ω ) and s α ( ω ) are nolonger in relation with the structure of the dendrogram‘cut’ at a given point in the vertical dimension, since thelatter does not represent the original distances, due tothe lack of ultrametricity. Appendix E: The modified Cont-Bouchaud model
In its simplest formulation, the Cont-Bouchaud (CB)model [9] considers a population of individuals (in thefinancial jargon, agents) that can make a binary choicebetween buying or selling an asset traded in the market.We can represent the choice expressed by the i -th agentas φ i = ±
1. Binary choices are the simplest possibil-ity considered also in other models of social processes,such as voting dynamics [2]. In the CB model, the ef-fects of mutual influence are modeled by introducing arandom graph through which agents can exchange in-formation before making their choices. As a result ofthis information exchange, all the agents belonging tothe same connected component are assumed to collec-tively agree on the choice to make. Therefore, if A labelsa connected component of the graph, the choice of allagents belonging to A is the same ( φ i = φ A ∀ i ∈ A ),while different connected components make statisticallyindependent choices. The key result is that the probabil-ity distribution of the aggregate choice of all individuals(which in the CB model is the aggregate demand deter-mining the price change of the asset) crucially dependson the topology of the interaction graph. In particular,if the connection probability p is set at the critical value p c ∼ N − (for N → ∞ ) of the phase transition givingrise to the giant connected component, the distributionof the sizes of connected components acquires a power-law form, which in turn implies a power-law distributionof price returns similar to the empirically observed ones.Other values of p yield different outcomes. In the limit p = 0 (empty graph) all agents make independent choicesand the distribution becomes Gaussian. By contrast, for p = 1 (complete graph) all agents always make the samechoice and the distribution is double-peaked.In order to study the effects of the nontrivial distribu-tion of individuals in cultural space, we extended the CBmodel to a more general ‘coordination model’ which in-corporates a dependence on real data. In particular, weintroduce a more realistic mechanism allowing culturallysimilar agents to express similar preferences. To takethis aspect into account we assume that each agent is de-scribed by a cultural vector { ~v i } and that agents interact,rather than on a random graph defined by a value of p ,on the cultural graph defined by a value of the confidence ω . Again, agents within the same connected componentare assumed to make collectively the same choice, while2agents belonging to different components express statis-tically independent preferences.The overall outcome of the process (e.g. the result ofthe survey/referendum/election) is the sum of individualpreferences, and can be quantified by the average choiceΦ = 1 N N X i =1 φ i (E1)whose sign reflects the choice of the majority. If thechoices of all agents are independent ( ω = 0), Φ is thesum of N uncorrelated random variables with finite vari-ance and, as follows from the Central Limit Theorem,normally distributed. In such a case, if the individual bi-nary probabilities are equal (i.e. the events φ i = +1 and φ i = − P ω (Φ)that the average choice takes the particular value Φ issymmetric about the most probable value Φ = 0. If ω >
0, Φ can be rewritten as the sum over different con-nected components:Φ = 1 N X A S A φ A (E2)where now A labels the components, φ A = ± A , and S A is the sizeof A . Now, a crucial point is that even if each connectedcomponent makes one of the two choices φ A = ± P ω (Φ) isstill symmetric about Φ = 0, the symmetry breaks spon-taneously at the critical threshold ω c . To see this, onecan compute P ω (Φ) in the following manner. For eachgroup of individuals in our data, and for a given valueof ω (from ω = 0 to ω = 1 in increments of 0 . P ω (Φ) as thenormalized histogram of the values obtained. The aboveanalysis provides a method to determine with small in-determinacy (∆ ω = 0 .
01) the threshold value ω c for eachparticular, finite group under study. We repeated ouranalysis on each of the 13 sampled groups (real and ran-domized) and obtained the corresponding critical thresh-olds ω c ± ∆ ω .We found that, while the critical thresholds obtainedfor shuffled data (and, trivially, also random ones) areconsistent with each other, real data feature different crit-ical values. This implies that even two randomly sampledsocial groups (for instance one in Italy and one in Por-tugal) with the same size and under the same level ofmutual influence may evolve to opposite collective states(coordination or heterogeneity) if the critical thresholdsfor the two groups differ.We now prove rigorously the equality C ( ω ) ≡ σ ω (Φ) = vuutX A (cid:18) S A N (cid:19) ω (E3) which establishes a tight relation between network topol-ogy and the level of collective social behavior in themodel. If, for a given value of ω , we denote the expectedvalue of Φ as h Φ i ω = P Φ P ω (Φ)Φ and its second momentas h Φ i ω = P Φ P ω (Φ)Φ , the variance σ ω (Φ) is definedas σ ω (Φ) ≡ h Φ i ω − h Φ i ω (E4)For simplicity, in what follows we drop the dependence ofall quantities on ω . For a fixed value of ω , the sizes of theconnected components of the network are given by { S A } ,and determine the aggregate choice Φ through eq.(E2).Since Φ is a sum of the random variables { S A φ A /N } , itsvariance σ (Φ) can be easily expressed as σ (Φ) = X A X B S A S B N σ AB = X A S A σ A N + X A X B = A S A S B N σ AB (E5)where σ AB denotes the covariance between the choices φ A and φ B of two different connected components A and B , and σ A is the variance of φ A . Now, since σ A ≡ h φ A i − h φ A i = 1 (E6)and since different connected components make statisti-cally independent choices, it follows that σ AB ≡ h φ A φ B i − h φ A ih φ B i = δ AB σ A = δ AB (E7)where δ AB = 1 if A = B and δ AB = 0 if A = B . Thusthe variance of Φ is simply σ (Φ) = X A S A N (E8)which proves the last equality in eq.(E3).Note that for ω = 0 there are N connected componentsof size S A = 1 (all vertices are isolated) and therefore σ (Φ) = 1 /N (the results of the Central Limit Theoremare recovered). In the opposite limit ω = 1, the net-work is a single connected component of size S A = N ,which yields σ (Φ) = 1. Thus the social coordination C ( ω ) ≡ σ ω (Φ) varies from C (0) = 1 / √ N (no collectivebehavior) to C (1) = 1 (perfect collective behavior). Forgeneric values of ω note that, since P A S A = N , theexpression for σ (Φ) in eq.(E8) has the form of an in-verse participation ratio. This means that σ (Φ) ≃ /n if the sum over A is dominated by n terms of approx-imately equal size. In particular, σ (Φ) ≃ σ (Φ) ≃ /N if each connected component trivially contains only onevertex. This result rephrases the connection betweenthe shape of P (Φ) and the underlying network topol-ogy: when there is no giant component, the width of P (Φ) is σ (Φ) ≃ / √ N →
0, while when the giant com-ponent is there the width of P (Φ) has the finite value σ (Φ) ≃
1. Thus when there is no giant component theremust be a single peak, while the presence of two peaks3at finite distance necessarily implies the presence of thegiant component. In such a case, σ (Φ) gives an estimateof the separation between the peaks.Importantly, these results are valid as N goes to in-finity, therefore our method to compute ω c as the valuemarking a spontaneous symmetry breaking (from single-peaked to double-peaked) in the probability P ω (Φ) pro-vides a consistent way to define a ‘critical’ value, whichtechnically is defined only for infinite systems, even forour inherently finite data. For infinite systems, ourmethod would yield the correct value of the criticalthreshold. Other finite-size techniques would require as-sumptions about how topological quantities scale withnetwork size. While for theoretical models (such as theErd˝os-R´enyi random graph [26]) it is possible to derivethese assumptions, for real systems this is not possible. Appendix F: The modified Axelrod model
In the original version of the Axelrod model, N indi-viduals (in social science jargon, ‘actors’) sitting at thevertices of a social network are represented as vectorsof cultural traits (or features), that evolve through dis-crete steps. In an elementary time-step, an individual i and one of his neighbors (say j ) are selected. Then thenormalized overlap o ij ∈ [0 ,
1] between their cultural vec-tors is computed as the fraction of identical components(note that the overlap is related to the cultural distance d ij through o ij = 1 − d ij ). With probability equal to o ij , the two actors interact: one of j ’s traits, chosen ran-domly among the set of traits where i and j differ, ischanged and set equal to the corresponding trait of i .Otherwise nothing happens, and two other actors are se-lected. These rules implement the two basic mechanismsof social influence (neighboring actors tend to convergeculturally) and homophily (similar individuals interactmore frequently). The Axelrod model leads to the im-portant conclusion that these two mechanisms do notnecessarily reinforce each other leading to a culturallyhomogeneous society. In fact, the model predicts thatdiversity is preserved: when two individuals become com-pletely different (zero overlap), they no longer interact.Thus in the allowed final configurations two neighboringactors are either completely identical or completely dif-ferent, and the society is split into cultural domains ofidentical vectors, with no overlap between adjacent do-mains. The average h N D i (over many realizations) ofthe number N D of different domains in the final stage, orequivalently the fraction h N D i /N , is a convenient way tomeasure the predicted cultural diversity as a function ofthe model parameters.As for the CB model, for our purposes it is impor-tant to incorporate real data into the Axelrod model. Inthis case, convenient generalizations have already beenproposed. While in the original model traits were non-metric, a modification bringing us closer to real data isthe introduction of metric features and the consequent ‰‰‰‰ ‰ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ ‰+++++++++++++++++++ +ƒ ƒ ƒ ƒƒƒƒƒƒƒƒƒ ƒ a L ‰ Real + Shuffled ƒ Randoma0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70100200300400500 Ω X N D \ ƒƒƒƒƒƒƒƒƒƒƒƒƒ +++++++++++++++++++++ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ b L ‰ Real + Shuffled ƒ Randoma0 100 200 300 400 5000100200300400500 N C X N D \ FIG. 8: The ultrametric properties of real opinions constrainthe evolution of cultural convergence even for infinite-rangeinteractions in social space. a) The average number h N D i ω of cultural domains obtained in the final state of the Axelrodmodel as a function of ω , when the initial state is given byreal, shuffled, and random opinions. b) The average number h N D i of final cultural domains versus the number N C of initialconnected components in the cultural graphs, for real, shuffledand random opinions. redefinition of cultural distance d ij as a metric distancebetween cultural vectors [18]. Another important vari-ant introduces the effect of bounded confidence: Flacheand Macy [17] introduced a threshold θ such that, ifthe overlap is smaller than or equal to θ , no interactiontakes place. Otherwise, it takes place with probability o ij (the original version of the model is recovered if θ = 0).Clearly, θ has exactly the same meaning as 1 − ω , where ω is the confidence we introduced above. The thresholdcompensates the effect that, as the number of featuresgrows, the presence of completely different pairs of agentsbecomes unlikely. It also allows to reproduce more realis-tically the fact that individuals are uninfluenced by eachother if they are different enough, not necessarily on eachand every opinion they have. [34]The above modifications allowed us to use the empiri-cal cultural vectors (rather than commonly assumed uni-formly random vectors) as the starting configuration, andstudy the final diversity predicted by the model. In orderto simulate long-range online-like dynamics we assumedthat the ‘social’ network is a complete graph. Since we4never found more than one individual with exactly thesame cultural vector, the initial number of cultural do-mains was always N = 500 for each sampled group. Infig.8a we report the average number h N D i ω of differentcultural domains in the final state of the dynamics, asa function of ω = 1 − θ for real, shuffled and randomopinions. As can be seen, for large values of θ (small ω )the final fraction of culturally homogeneous domains isfinite (at the extreme θ = 1 there is no evolution from thestarting configuration and the initial vectors remain alldistinct), while for small values of θ (large ω ) the samefraction is of order 1 /N (and vanishes at the extreme θ = 0 corresponding to the ordinary Axelrod model).This means that the final cultural diversity decreases as ω increases. With respect to real data, the curved for shuf-fled and random data are moved rightwards. Naively, ifcombined with our previous results, this finding appearsto confirm the expectation that larger cultural diversity isonly reached in a regime (small ω ) where short-term col-lective behavior is weak or absent, and conversely strong collective behavior can only exist for large values of ω which suppress cultural heterogeneity in the long run.Moreover, this appears to apply equally to real, shuffledand random data.However, this conclusion is incorrect. In fig.8b we showthe average number h N D i of final different cultural do-mains versus the number N C of initial connected compo-nents in the underlying cultural network, both obtainedfor various values of the threshold ω and for the threeusual cases of real, shuffled and random data. We findthat, for a given value of N C , real data are those thatachieve the largest level of long-term cultural heterogene-ity (value of h N D i ). Indeed, for real data the realizedvalue of h N D i is the largest possible[35] ( h N D i ≈ N C ) in-dicating that cultural convergence is confined within theinitial connected components, each of which eventuallybecomes a single cultural domain. By contrast, in ran-domized data there are less final cultural domains thaninitial connected components, indicating that the latteroften ‘merge’ into larger cultural domains. [1] M. Buchanan, The social atom (Bloomsbury, 2007).[2] C. Castellano, S. Fortunato, and V. Loreto, Reviews ofmodern physics , 591 (2009).[3] D. Lazer, A. Pentland, L. Adamic, S. Aral, A.-L.Barabasi, D. Brewer, N. Christakis, N. Contractor,J. Fowler, M. Gutmann, et al., Science , 721 (2009).[4] S. Fortunato and C. Castellano, Physical review letters , 138701 (2007).[5] J. Fowler and N. Christakis, Proceedings of the NationalAcademy of Sciences , 5334 (2010).[6] J. Ratkiewicz, S. Fortunato, A. Flammini, F. Menczer,and A. Vespignani, Phys. Rev. Lett. , 158701 (2010).[7] J.-P. Onnela and F. Reed-Tsochas, Proceedings of theNational Academy of Sciences , 18375 (2010).[8] R. Mantegna and H. Stanley, An introduction to econo-physics: correlations and complexity in finance (Cam-bridge Univ Pr, 2000).[9] R. Cont and J. Bouchaud, Macroeconomic Dynamics ,170 (2000).[10] S. Sinha, A. Chatterjee, A. Chakraborti, andB. Chakrabarti, Econophysics: an introduction (Wiley-VCH Verlag GmbH, 2010).[11] B. Chakrabarti, A. Chakraborti, and A. Chatterjee,