Individualization as driving force of clustering phenomena in humans
11 Individualization as driving force of clustering phenomena inhumans
Michael M¨as , ∗ , Andreas Flache , Dirk Helbing , , ∗ E-mail: [email protected]
Abstract
One of the most intriguing dynamics in biological systems is the emergence of clustering, the self-organization into separated agglomerations of individuals. Several theories have been developed to explainclustering in, for instance, multi-cellular organisms, ant colonies, bee hives, flocks of birds, schools of fish,and animal herds. A persistent puzzle, however, is clustering of opinions in human populations. Thepuzzle is particularly pressing if opinions vary continuously, such as the degree to which citizens are infavor of or against a vaccination program. Existing opinion formation models suggest that “monoculture”is unavoidable in the long run, unless subsets of the population are perfectly separated from each other.Yet, social diversity is a robust empirical phenomenon, although perfect separation is hardly possible inan increasingly connected world. Considering randomness did not overcome the theoretical shortcomingsso far. Small perturbations of individual opinions trigger social influence cascades that inevitably lead tomonoculture, while larger noise disrupts opinion clusters and results in rampant individualism withoutany social structure.Our solution of the puzzle builds on recent empirical research, combining the integrative tendenciesof social influence with the disintegrative effects of individualization. A key element of the new computa-tional model is an adaptive kind of noise. We conduct simulation experiments to demonstrate that withthis kind of noise, a third phase besides individualism and monoculture becomes possible, characterizedby the formation of metastable clusters with diversity between and consensus within clusters. When clus-ters are small, individualization tendencies are too weak to prohibit a fusion of clusters. When clustersgrow too large, however, individualization increases in strength, which promotes their splitting.In sum, the new model can explain cultural clustering in human societies. Strikingly, our modelpredictions are not only robust to “noise”—randomness is actually the central mechanism that sustainspluralism and clustering.
Introduction
Many biological systems exhibit collective patterns, which emerge through simple interactions of largenumbers of individuals. A typical example are agglomeration phenomena. Such clustering dynamics havebeen found in systems as different as bacterial colonies [1], gregarious animals like cockroaches [2], fishschools [3], flocks of birds [4], and animal groups [5]. Similar phenomena are observed in ecosystems [6]and human populations, as the formation of urban agglomerations demonstrates [7, 8].Recently, the clustering of social interactions in human populations [9–11] has been extensively studiedin networks of email communication [12], phone calls [11], scientific collaboration [13] and sexual contacts[14]. It is much less understood, however, how the pattern of opinion clustering that characterizes modernsocieties comes about. Empirical studies suggest that in modern societies opinions differ globally [15],while they cluster locally within geographical regions [17], socio-demographic groups [18], or internetcommunities [19]. The lack of a theoretical understanding of opinion clustering is pressing, since both, a r X i v : . [ phy s i c s . s o c - ph ] J u l local consensus and global diversity are precarious. On the one hand, cultural diversity may get lost ina world where people are increasingly exposed to influences from mass media, Internet communication,interregional migration, and mass tourism, which may promote a universal monoculture [20, 21], asthe extinction of languages suggests [22]. On the other hand, increasing individualization threatens todisintegrate the social structures in which individuals are embedded, with the possible consequence of theloss of societal consensus [23,24]. This is illustrated by the decline of the social capital binding individualsinto local communities [25].Early formal models of social influence imply that monoculture is unavoidable, unless a subset of thepopulation is perfectly cut off from outside influences [26]. Social isolation, however, appears questionableas explanation of pluralism. In modern societies, distances in social networks are quite short on the whole,and only relatively few random links are required to dramatically reduce network distance [9].Aiming to explain pluralism, researchers have incorporated the empirically well-supported observationof “homophily”, i.e. the tendency of “birds of a feather to flock together” [27, 28], into formal models ofsocial influence [29]. These models typically assume “bounded confidence” (BC) in the sense that onlythose individuals interact, whose opinions do not differ more than a given threshold level [30, 31]. AsFig. 1A illustrates, BC generates opinion clustering, a result that generalizes to model variants withcategorical rather than continuous opinions [29, 32]. However, clustering in the BC-model is sensitive to“interaction noise”: A small random chance that agents may interact even when their opinions are notsimilar, causes monoculture again (see Fig. 1B).To avoid this convergence of opinions, it was suggested that individuals would separate themselvesfrom negatively evaluated others [18,33]. However, recent empirical results do not support such “negativeinfluence” [34]. Scientists also tried to avoid convergence by “opinion noise”, i.e. random influences, whichlead to arbitrary opinion changes with a small probability. Assuming uniformly distributed opinionnoise [35] leads to sudden, large, and unmotivated opinion changes of individuals, while theories ofsocial integration [23, 24, 36, 37] and empirical studies of individualization [38, 39] show a tendency ofincremental opinion changes rather than arbitrary opinion jumps. Incremental opinion changes, however,tend to promote monoculture, even in models with categorical rather than continuous opinions [40].Figure 1 demonstrates that adding a “white noise” term ( N (0 , θ )) to an agent’s current opinion in theBC model fails to explain opinion clustering. Weak opinion noise ( θ = 5) triggers convergence cascadesthat inevitably end in monoculture. Stronger noise restores opinion diversity, but not pluralism. Instead,diversity is based on frequent individual deviations from a predominant opinion cluster (for θ = 18).However, additional clusters can not form and persist, because opinion noise needs to be strong toseparate enough agents from the majority cluster—so strong that randomly emerging smaller clusterscannot stabilize.In conclusion, the formation of persistent opinion clusters is such a difficult puzzle that all attemptsto explain them had to make assumptions that are difficult to justify by empirical evidence. The solutionproposed in the following, in contrast, builds on sociological and psychological research. The key innova-tion is to integrate another decisive feature into the model, namely the “strive for uniqueness” [38, 39].While individuals are influenced by their social environment, they also show a desire to increase theuniqueness when too many other members of society hold similar opinions. This suggests that noise inindividual opinion formation is adaptive, creating a dynamic interplay of the integrating and disinte-grating forces highlighted by Durkheim’s classic theory of social integration [23]. Durkheim argued thatintegrating forces bind individuals to society, motivating them to conform and adopt values and normsthat are similar to those of others. But he also saw societal integration as being threatened by disinte-grating forces that foster individualization and drive actors to differentiate from one another [24, 36, 37].The continuous “Durkheimian opinion dynamics model” proposed in the following can explain pluralism,although it incorporates all the features that have previously been found to undermine clustering: (1)a fully connected influence network, (2) absence of bounded confidence, (3) no negative influence, and(4) white opinion noise. From a methodological viewpoint, our model builds on concepts from statisticalphysics, namely the phenomenon of “nucleation” [41], illustrated by the formation of water droplets insupersaturated vapor. However, by assuming adaptive noise, we move beyond conventional nucleationmodels.Computational experiments reveal that our model generates pluralism as an intermediate phase be-tween monoculture and individualism. When the integrating forces are too strong, the model dynamicsinevitably implies monoculture, even when the individual opinions are initially distributed at random.When the disintegrating forces prevail, the result is what Durkheim called “anomie”, a state of extremeindividualism without a social structure, even if there is perfect consensus in the beginning. Interest-ingly, there is no sharp transition between these two phases, when the relative strength of both forces ischanged. Instead, we observe an additional, intermediate regime, where opinion clustering occurs, whichis independent of the initial condition. In this regime, adaptive noise entails robust pluralism that isstabilized by the adaptiveness of cluster size. When clusters are small, individualization tendencies aretoo weak to prohibit a fusion of clusters. However, when clusters grow large, individualization increasesin strength, which triggers a splitting into smaller clusters (“fission”). In this way, our model solves thecluster formation problem of earlier models. While in BC models, white noise causes either monocul-ture or fragmentation (Fig. 1C), in the Durkheimian opinion dynamics model proposed here, it enables clustering. Therefore, rather than endangering cluster formation, noise supports it. In the following, wedescribe the model and identify conditions under which pluralism can flourish. Model
The model has been elaborated as an agent-based model [42] addressing the opinion dynamics of in-teracting individuals. The simulated population consists of N agents i , representing individuals, eachcharacterized by an opinion o i ( t ) at time t . The numerical value for the opinion varies between a givenminimum and maximum value on a metric scale. We use the term “opinion” here, for consistency withthe literature on social influence models. However, o i may also reflect behaviors, beliefs, norms, customsor any other cardinal cultural attribute that individuals consider relevant and that is changed by socialinfluence. The dynamics is modeled as a sequence of events. Every time t (cid:48) = k/N (with k ∈ { , ..., N } ),the computer randomly picks an agent i and changes the opinion o i by the amount∆ o i = N (cid:88) j =1 j (cid:54) = i (cid:0) o j − o i (cid:1) w ijN (cid:88) j =1 j (cid:54) = i w ij + ξ i . (1)The first term on the rhs of Eq. [1] models the integrating forces of Durkheim’s theory. Technically,agents tend to adopt the weighted average of the opinions o j of all other members j of the population.Implementing homophily, the social influence w ij that agent j has on agent i is the stronger, the smallertheir opinion distance d ij = | o j − o i | is. Formally, we assume w ij = e − d ij /A = e −| o j − o i | /A . (2)The parameter A represents the range of social influence of agents. For small positive values of A , agentsare very confident in their current opinion and are mainly influenced by individuals who hold very similaropinions, while markedly distinct opinions have little impact. The higher A is, however, the more areagents influenced by individuals with considerably different opinions and the stronger are the integratingforces in our Durkheimian theory.The dis integrating forces on the opinion of agent i are modeled by a noise term ξ i . Specifically, thecomputer adds a normally distributed random value ξ i (“white noise”) to the first term on the rhs of Eq.[1]. While we assume that the mean value of the random variable ξ i is zero, the standard deviation hasbeen specified as θ it = s N (cid:88) j =1 e − d ij . (3)The larger the standard deviation, the stronger are the individualization tendencies of an agent.Following Durkheim’s theory, equation [3] implements the assumption that an agent’s strive for individ-ualization is weak, if there are only a few others with similar opinions. Under such conditions, there isno need to increase distinctiveness. However, if many others hold a similar opinion, then individuals aremore motivated to differ from others.By including the focal agent i in the sum of Eq. [3], we assume that there is always some opinionnoise, even when agent i holds a perfectly unique opinion. These fluctuations may have a variety ofreasons, such as misjudgements, trial-and-error behavior, or the influence of exogenous factors on theindividual opinion.We use the parameter s to vary the strength of the disintegrating forces in society. The higher thevalue of s , the higher is the standard deviation of the distribution, from which ξ i is drawn, and thestronger are the disintegrating forces. Finally, to keep the opinions of the agents within the bounds of theopinion scale, we set the value of ξ i to zero, if the bounds of the opinion space would be left otherwise. Results
We have studied the Durkheimian opinion dynamics model with extensive computer simulations, focussingourselves on relatively small populations ( N = 100), because in this case it is reasonable to assume thatall members may interact with each other. For bigger groups one would have to take into account thetopology of the social interaction network as well. Such networks would most likely consist of segregatedcomponents (“communities”), which are not or only loosely connected with each other [11–14]. Becauseof the weak or missing connections between communities, it would not be so surprising if each communitydeveloped its own, shared opinion. In small, completely connected populations, however, the occurrenceof diverse opinions is puzzling, as it cannot result from a lack of contacts between agents.To illustrate the model dynamics, Fig. 2 shows three typical simulation runs for different strengths s of disintegrating forces, while the strength A = 2 of the integrating force is kept constant. In each run, allagents start with an opinion in the middle of the opinion scale ( o i = 0), i.e. conformity. This is an initialcondition for which the classical BC-model does not produce diversity. Fig. 2A shows typical opiniontrajectories for a population, in which the integrating forces are much stronger than the disintegratingforces. Consequently, the population develops collective consensus, i.e. the variation of opinions remainssmall, even though not all agents hold exactly the same opinion. Triggered by the random influences ξ i ,the average opinion performs a characteristic random walk.When the disintegrating force prevails, the pattern is strikingly different. Fig. 2B shows that for largenoise strengths s , the initial consensus breaks up quickly, and the agents’ opinions are soon scatteredacross the entire opinion space.Simulation scenarios A and B are characteristic for what Durkheim referred to as states of socialcohesion and of anomie. Interestingly, however, pluralism arises as a third state in which several opinionclusters form and coexist. Fig. 2C shows a typical simulation run, where the adaptive noise maintainspluralism depite the antagonistic impacts of integrating and disintegrating forces—in fact because of this.In the related region of the parameter space, disintegrating forces prevent global consensus, but theintegrating forces are strong enough to prevent the population from extreme individualization. This is inpronounced contrast to what we found for the BC-model with strong noise (Fig. 1C). Instead, we obtaina number of coexisting, metastable clusters of a characteristic, parameter-dependent size. Each clusterconsists of a relatively small number of agents, which keeps the disintegrating forces in the cluster weakand allows clusters to persist. (Remember that the tendency of individualization according to Eq. [3]increases, when many individuals hold similar opinions.) However, due to opinion drift, distinct clustersmay eventually merge. When this happens, the emergent cluster becomes unstable and will eventuallysplit up into smaller clusters, because disintegrating forces increase in strength as a cluster grows.Strikingly, the state of diversity, in which several opinion clusters can coexist, is not restricted to anarrow set of conditions under which integrating and disintegrating forces are balanced exactly. Fig. 3demonstrates that opinion clusters exist in a significant area of the parameter space, i.e. the clusteringstate establishes another phase, which is to be distinguished from monoculture and from anomie.To generate Fig. 3, we conducted a simulation experiment in which we varied the influence range A and the strength s of the disintegrating force. For each parameter combination, we ran 100 replicationsand measured the average number of clusters that were present after 250,000 iterations. To count thenumber of clusters in a population, we ordered the N agents according to their opinion. A cluster wasdefined as a set of agents in adjacent positions such that each set member was separated from the adjacentset members by a maximum of 5 scale points (= opinion range/ N ). Fig. 3 shows that, for large socialinfluence ranges A and small noise strengths s , the average number of clusters is below 1.5, reflectingmonoculture in the population. In the other extreme, i.e. for a small influence range A and large noisestrengths s , the resulting distribution contains more than 31 clusters, a number of clusters that cannotbe distinguished from purely random distributions. Following Durkheim, we have classified such casesas anomie, i.e. as the state of extreme individualism. Between these two phases, there are numerousparameter combinations, for which the number of clusters is higher than 1.5 and clearly smaller thanin the anomie phase. This constitutes the clustering phase. Fig. 3 also shows that, for each parametercombination, there is a small variance in the number of clusters, which is due to a statistical equilibriumof occasional fusion and fission processes of opinion clusters (see Fig. 2C).The same results were found, when starting the computer simulations with a uniform opinion dis-tribution. Additional statistical tests were performed to make sure that the existence of clusters in ourmodel indeed indicates pluralism and not fragmentation, a state in which a population consists of one bigcluster and a number of isolated agents (see Fig. 4). While the Durkheimian opinion dynamics model isconsistent with cluster formation (see Fig. 4A), the noisy BC model rather shows random fragmentation(see Fig. 4B). Discussion
The phenomenon of self-organized clustering phenomena in biological and social systems is widespreadand important. With the advent of mathematical and computer models for such phenomena, there hasbeen an increasing interest to study them also in human populations. The work presented here focuseson resolving the long-standing puzzle of opinion clustering.The emergence and persistence of pluralism is a striking phenomenon in a world in which socialnetworks are highly connected and social influence is an ever present force that reduces differences betweenthose who interact. We have developed a formal theory of social influence that, besides anomie andmonoculture, shows a third, pluralistic phase characterized by opinion clustering. It occurs, when allindividuals interact with each other and noise prevents the convergence to a single opinion, despitehomophily.Our model does not assume negative influence, and it behaves markedly different from boundedconfidence models, in which white opinion noise produces fragmentation rather than clustering. It wouldbe natural to generalize the model in a way that also considers the structure of real social networks.This basically requires one to replace the values w ij by w ij a ij , where a ij are the entries of the adjacencymatrix (i.e. a ij = 1, if individuals i and j interact, otherwise a ij = 0). In such a case, resulting opinionclusters are expected to have a broad range of different sizes, similar to what is observed for the sizes ofsocial groups.Our model highlights the functional role that “noise” (randomness, fluctuations, or other sources ofvariability) plays for the organization of social systems. It furthermore shows that the combination of twomechanisms (deterministic integrating forces and stochastic disintegrating forces) can give rise to newphenomena. We also believe that our results are meaningful for the analysis of the social integration of oursocieties. Both classical [23] and contemporary [24] social thinkers argue that, in modern and globalizedsocieties, individuals are increasingly exposed to disintegrating forces that detach them from traditionalsocial structures. In other words, the social forces that motivate individuals to follow societal norms maylose their power to limit individual variation. Durkheim feared that this will atomize societies [23]. Thatis, he thought the tendency towards individualization would turn societies “anomic” as they modernize:Durkheim felt that extreme individualization in modern societies would obstruct the social structuresthat traditionally provided social support and guidance to individuals.Today, modern societies are highly diverse, but at the same time they are far from a state of anomie asforeseen by Durkheim. Our model offers an explanation why and how this is possible: Besides monocultureand anomie, there is a third, pluralistic clustering phase, in which individualization prevents overallconsensus, but at the same time, social influence can still prevent extreme individualism. The interplaybetween integrating and disintegrating forces leads to a plurality of opinions, while metastable subgroupsoccur, within which individuals find a local consensus. Individuals may identify with such subgroups anddevelop long-lasting social relationships with similar others. Therefore, they are not isolated and notwithout support or guidance, in contrast to the state of anomie that Durkheim was worried about.We have seen, however, that pluralism and cultural diversity require an approximate balance betweenintegrating and disintegrating forces. If this balance is disturbed, societies may drift towards anomie ormonoculture. It is, therefore, interesting to ask how the current tendency of globalization will influencesociety and cultural dynamics. The Internet, interregional migration, and global tourism, for example,make it easy to get in contact with members of distant and different cultures. Previous models [21, 32]suggest that this could affect cultural diversity in favor of a monoculture. However, if the individual strivefor uniqueness is sufficiently strong, formation of diverse groups (a large variety of international socialcommunities) should be able to persist even in a globalizing world. In view of the alternative futures,characterized by monoculture or pluralism, further theoretical, empirical, and experimental researchshould be performed to expand our knowledge of the mechanisms that will determine the future ofpluralistic societies. References
1. Ben-Jacob E, Schochet O, Tenenbaum A, Cohen I, Czirok A, Vicsek T (1994) Generic Modelingof Cooperative Growth-Patterns in Bacterial Colonies.
Nature
Anim Behav
69: 169-180.3. Gautrais J, Jost C, Theraulaz G (2008) Key behavioural factors in a self-organised fish schoolmodel.
Ann Zool Fenn
45: 415-428.4. Ballerini M, Calbibbo N, Candeleir R, Cavagna A, Cisbani E et al. (2008) Interaction ruling animalcollective behavior depends on topological rather than metric distance: Evidence from a field study.
Proc Natl Acad Sci USA
Nature
Ecol Model
37: 287-302.7. Makse HA, Havlin S, Stanley HE (1995) Modeling Urban-Growth Patterns.
Nature
Science
Nature
Science
Nature
Proc Natl Acad Sci USA
Proc NatlAcad Sci USA
Nature
Annu Rev Polit Sci
11: 563-588.16. DiMaggio P, Evans J,Bryson B (1996) Have Americans’ social attitudes become more polarized?
Am J Sociol
J Econ Perspect
20: 119-144.18. Mark NP (2003) Culture and competition: Homophily and distancing explanations for culturalniches.
Am Sociol Rev
68: 319-345.19. Lazer D et al. (2009) Computational Social Science.
Science 323 : 721-723.20. Friedman TL (2005) The World is Flat. A brief history of the twenty-first century (Farrar, Strausand Giroux, New York).21. Greig MJ (2002) The end of geography? Globalization, communication and culture in the interna-tional system.
J Conflict Res
46: 225-243.22. Sutherland WJ (2003) Parallel extinction risk and global distribution of languages and species.
Nature
The Division of Labor in Society (The Free Press, New York).24. Beck U (1994) in
Reflexive Modernization. Politics, Tradition and Aestherics in the Modern SocialOrder , eds. Beck U, Giddens A, Lash S (Polity Press, Cambridge)25. McPherson M, Smith-Lovin L,Brashears ME (2006) Social isolation in America: Changes in corediscussion networks over two decades.
Am Sociol Rev
71: 353-375; also consult the discussion onthis article in
Am Sociol Rev : 74:4.26. Abelson RP (1964) in
Contributions to Mathematical Psychology , eds. Frederiksen N, Gulliksen H(Rinehart Winston, New York), pp 142-160.27. McPherson M, Smith-Lovin L, Cook JM (2001) Birds of a feather: Homophily in social networks.
Annu Rev Sociol
Proc Natl Acad Sci USA
Psychol Rev
97: 362-376.30. Hegselmann R, Krause U (2002) Opinion dynamics and bounded confidence models. Analysis, andSimulation
J Artif Soc S
Am J Soc 110 : 1041-1069.32. Axelrod R (1997) The dissemination of culture - A model with local convergence and global polar-ization.
J Conflict Res
41: 203-226.33. Macy MW, Kitts J, Flache A, Benard S (2003) in
Dynamic Social Network Modelling and Analysis ,eds. Breiger R, Carley K, Pattison P (The National Academies Press, Washington, DC), pp. 162-173.34. Krizan Z, Baron RS (2007) Group polarization and choice-dilemmas: How important is self-categorization?
Eur J Soc Psychol
37: 191-201.35. Pineda M, Toral R, Hernandez-Garcia E (2009) Noisy continuous-opinion dynamics.
Journal ofStatistical Mechanics : P08001.36. Hornsey MJ, Jetten J (2004) The individual within the group: Balancing the need to belong withthe need to be different.
Pers Soc Psychol Rev
8: 248-264.37. Vignoles VL, Chryssochoou X, Breakwell GM (2000) The distinctiveness principle: Identity, mean-ing, and the bounds of cultural relativity.
Pers Soc Psychol Rev
4: 337-354.38. Imhoff R, Erb HP (2009) What motivates nonconformity? Uniqueness seeking blocks majorityinfluence.
Pers Soc Psychol B
35: 309-320.39. Snyder CR, Fromkin HL (1980)
Uniqueness. The Human Pursuit of Difference (Plenum Press, NewYork and London).40. Klemm K, Eguiluz VM, Toral R, Miguel MS (2003) Global culture: A noise-induced transition infinite systems.
Phys Rev E
67: 045101(R).41. Stanley, HE (1971)
Introduction to Phase Transitions and Critical Phenomena (Oxford UniversityPress, Oxford and New York).42. Bonabeau E (2002) Agent-based modeling: Methods and techniques for simulating human systems.
Proc Natl Acad Sci USA
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Figure 1. Opinion dynamics produced by the bounded confidence (BC) model [30] withand without noise . Populations consist of 100 agents. Opinions vary between -250 and 250. Initialopinions are uniformly distributed. For visualization, the opinion scale is divided into 50 bins of equalsize. Color coding indicates the relative frequency of agents in each bin. (A) Dynamics of the
BC-modelwithout noise [30] over 10 iterations. At each simulation event, one agent’s opinion is replaced by theaverage opinion of those other agents who hold opinions o j within the focal agent’s confidence interval( o i − (cid:15) ≤ o j ≤ o i + (cid:15) ). For (cid:15) = 0 .
05, one finds several homogeneous clusters, which stabilize when thedistance between all clusters exceeds the confidence threshold (cid:15) . (B) Computer simulation of the sameBC-model, but considering interaction noise . Agents that would otherwise not have been influential,now influence the focal agent’s opinion with a probability of p = 0 .
01. This small noise is sufficient toeventually generate monoculture. (C) Simulation of the BC-model with opinion noise. After eachopinion update, a normally distributed random value drawn from N (0 , θ ) is added to the opinion.Under weak opinion noise ( θ = 5), one cluster is formed, which carries out a random walk on theopinion scale. When the opinion noise is significantly increased ( θ = 18), there is still one big cluster,but many separated agents exist as well (cf. Fig. 4). With even stronger opinion noise ( θ = 20), theopinion distribution becomes completely random.0 (cid:111)(cid:112)(cid:110) (cid:105)(cid:105) (cid:111)(cid:110) (cid:45)(cid:50)(cid:53)(cid:48)(cid:50)(cid:53)(cid:48)(cid:48) (cid:115)(cid:61)(cid:46)(cid:52) (cid:48) (cid:48)(cid:49) (cid:48) (cid:48)(cid:48) (cid:48)(cid:57) (cid:48)(cid:48) (cid:48)(cid:56) (cid:48)(cid:48) (cid:48)(cid:55) (cid:48)(cid:54) (cid:48) (cid:48) (cid:48)(cid:53) (cid:48) (cid:48) (cid:48)(cid:52) (cid:48) (cid:48) (cid:48)(cid:51) (cid:48) (cid:48) (cid:48)(cid:50) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:49) (cid:48)(cid:48) (cid:65)(cid:58)(cid:32) (cid:77)(cid:111)(cid:110)(cid:111)(cid:99)(cid:117)(cid:108)(cid:116)(cid:117)(cid:114)(cid:101) (cid:112)(cid:110)(cid:111) (cid:105)(cid:105) (cid:111)(cid:110) (cid:45)(cid:50)(cid:53)(cid:48)(cid:50)(cid:53)(cid:48)(cid:48) (cid:49) (cid:48) (cid:48) (cid:48) (cid:48)(cid:57) (cid:48) (cid:48) (cid:48)(cid:56) (cid:48) (cid:48) (cid:48)(cid:55) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:54) (cid:48)(cid:48) (cid:48)(cid:53) (cid:48)(cid:48) (cid:48)(cid:52) (cid:48)(cid:51) (cid:48) (cid:48) (cid:48)(cid:50) (cid:48) (cid:48) (cid:48)(cid:49) (cid:48) (cid:48) (cid:48)(cid:48) (cid:66)(cid:58)(cid:32) (cid:65)(cid:110)(cid:111)(cid:109)(cid:105)(cid:101) (cid:49) (cid:48) (cid:48) (cid:48)(cid:48)(cid:57) (cid:48) (cid:48) (cid:48)(cid:56) (cid:48) (cid:48) (cid:48)(cid:55) (cid:48) (cid:48) (cid:48)(cid:54) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:53) (cid:48)(cid:48) (cid:48)(cid:52) (cid:48)(cid:48) (cid:48)(cid:51) (cid:48)(cid:50) (cid:48) (cid:48) (cid:48)(cid:49) (cid:48) (cid:48) (cid:48)(cid:48) (cid:67)(cid:58)(cid:32) (cid:67)(cid:108)(cid:117)(cid:115)(cid:116)(cid:101)(cid:114)(cid:105)(cid:110)(cid:103) (cid:111)(cid:112)(cid:110) (cid:105) (cid:110) (cid:105) (cid:111) (cid:45)(cid:50)(cid:53)(cid:48)(cid:50)(cid:53)(cid:48)(cid:48) (cid:115)(cid:61)(cid:49)(cid:46)(cid:50) (cid:112)(cid:110)(cid:111) (cid:105)(cid:105) (cid:111)(cid:110) (cid:45)(cid:50)(cid:53)(cid:48)(cid:50)(cid:53)(cid:48)(cid:48) (cid:50) (cid:52) (cid:54) (cid:56) (cid:49)(cid:48) (cid:49)(cid:50) (cid:49)(cid:52) (cid:49)(cid:54) (cid:49)(cid:56) (cid:50)(cid:48) (cid:50)(cid:50) (cid:50)(cid:52)(cid:46)(cid:46)(cid:46)(cid:48) (cid:78)(cid:117)(cid:109)(cid:98)(cid:101)(cid:114)(cid:32)(cid:111)(cid:102)(cid:32) (cid:97)(cid:103)(cid:101)(cid:110)(cid:116)(cid:115) Figure 2. Opinion trajectories of three representative simulation runs with 100 agentsgenerated by the Durkheimian model.
In all three runs, the opinions are restricted to valuesbetween -250 and 250, and all agents hold the same opinion initially ( o i (0) = 0 for all i ). In all runs, weassume the same social influence range A = 2, but vary the strength s of the disintegrating force. (A)Monoculture, resulting in the case of a weak disintegrating force ( s = 0 . s = 6). Agents spread overthe complete opinion scale. The black line represents the time-dependent opinion of a single, randomlypicked agent, showing significant opinion changes over time, which is in contrast to the collectiveopinion formation dynamics found in the monocultural and pluralistic cases (A) and (B). (C) For amoderate disintegrating force ( s = 1 . s = 1 .
2. It shows that the composition of clusters persists over long time periods.1
Figure 3. Conditions of clustering, monoculture and Anomie.
The figure shows thedependence of the average number of clusters in the Durkheimian model on the strength s of thedisintegrating force and the range A of social influence. To generate it, we conducted computersimulations with N = 100 agents, starting with initial consensus ( o i (0) = 0 for all i ). We restrictedopinions to values between -250 and 250. We varied the strength s of the disintegrating force between s = 0 . s = 8 in steps of 0.4. A varied between A = 0 . A = 4 in steps of 0.2. For eachparameter combination, we conducted 100 independent replications and assessed the average number ofclusters formed after 250,000 iterations (see z -axis and the color scale). The two transparent (gray)surfaces depict the inter-quartile range, which indicates a small variance in the number of clusters (andalso typical cluster sizes) for each parameter combination. The horizontal grids indicate the borders ofthe three phases, as defined by us. An average cluster size below 1.5 indicates monoculture. Valuesbetween 1.5 and 31 reflect clustering. Finally, values above 31 correspond to opinion distributions thatcannot be distinguished from random ones and represent a state of anomie.2 (cid:66)(cid:58)(cid:32) (cid:66)(cid:67)(cid:45)(cid:77)(cid:111)(cid:100)(cid:101)(cid:108)(cid:32) (cid:119)(cid:105)(cid:116)(cid:104)(cid:32) (cid:87)(cid:104)(cid:105)(cid:116)(cid:101)(cid:32) (cid:78)(cid:111)(cid:105)(cid:115)(cid:101)(cid:65)(cid:58)(cid:32) (cid:68)(cid:117)(cid:114)(cid:107)(cid:104)(cid:101)(cid:105)(cid:109)(cid:105)(cid:97)(cid:110)(cid:32) (cid:77)(cid:111)(cid:100)(cid:101)(cid:108) (cid:48)(cid:50)(cid:48)(cid:52)(cid:48)(cid:54)(cid:48)(cid:56)(cid:48)(cid:49)(cid:48)(cid:48) (cid:83) (cid:105) (cid:122) (cid:101) (cid:32) (cid:111) (cid:102) (cid:32) (cid:66) (cid:105) (cid:103)(cid:103)(cid:101) (cid:115) (cid:116) (cid:32) (cid:67) (cid:108) (cid:117) (cid:115) (cid:116) (cid:101) (cid:114) (cid:50) (cid:52) (cid:54) (cid:56) (cid:49)(cid:48) (cid:49)(cid:50) (cid:49)(cid:52) (cid:49)(cid:54) (cid:49)(cid:56) (cid:50)(cid:48) (cid:50)(cid:50) (cid:50)(cid:52) (cid:50)(cid:54) (cid:50)(cid:56) (cid:51)(cid:48)(cid:78)(cid:117)(cid:109)(cid:98)(cid:101)(cid:114)(cid:32) (cid:111)(cid:102)(cid:32) (cid:67)(cid:108)(cid:117)(cid:115)(cid:116)(cid:101)(cid:114)(cid:115) (cid:114)(cid:97)(cid:110)(cid:100)(cid:111)(cid:109)(cid:102)(cid:114)(cid:97)(cid:103)(cid:109)(cid:101)(cid:110)(cid:116)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110) (cid:109) (cid:111)(cid:100) (cid:32) (cid:114) (cid:117) (cid:108) (cid:116) (cid:101) (cid:108) (cid:101) (cid:115) (cid:48)(cid:50)(cid:48)(cid:52)(cid:48)(cid:54)(cid:48)(cid:56)(cid:48)(cid:49)(cid:48)(cid:48) (cid:83) (cid:105) (cid:122) (cid:101) (cid:32) (cid:111) (cid:102) (cid:32) (cid:66) (cid:105) (cid:103)(cid:103)(cid:101) (cid:115) (cid:116) (cid:32) (cid:67) (cid:108) (cid:117) (cid:115) (cid:116) (cid:101) (cid:114) (cid:50) (cid:52) (cid:54) (cid:56) (cid:49)(cid:48) (cid:49)(cid:50) (cid:49)(cid:52) (cid:49)(cid:54) (cid:49)(cid:56) (cid:50)(cid:48) (cid:50)(cid:50) (cid:50)(cid:52) (cid:50)(cid:54) (cid:50)(cid:56) (cid:51)(cid:48)(cid:78)(cid:117)(cid:109)(cid:98)(cid:101)(cid:114)(cid:32) (cid:111)(cid:102)(cid:32) (cid:67)(cid:108)(cid:117)(cid:115)(cid:116)(cid:101)(cid:114)(cid:115) (cid:109) (cid:111)(cid:100)(cid:101) (cid:108) (cid:114) (cid:101) (cid:115) (cid:117) (cid:108) (cid:116) (cid:32) (cid:114)(cid:97)(cid:110)(cid:100)(cid:111)(cid:109)(cid:102)(cid:114)(cid:97)(cid:103)(cid:109)(cid:101)(cid:110)(cid:116)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110) Figure 4. Comparison of the (A) Durkheimian model and (B) the noisy BC-model.
Figuresshow the size of the biggest cluster over the number of clusters (in all simulation runs that resulted inmore than one and less than 32 clusters). Fig. 4A is based on the simulation experiment underlyingFigure 3. Figure 4B was generated by an experiment with the BC-model [30], in which we varied thebounded-confidence level (cid:15) between 0.01 and 0.15 in steps of 0.02 and the noise level θ between 5 and 50in steps of 5. We conducted 100 replications per parameter combination and measured the number ofclusters and the size of the biggest cluster after 250,000 iterations. White solid lines represent theaverage size of the biggest cluster. The dark blue area shows the respective interquartile range and thelight blue area the complete value range. For comparison, we generated randomly fragmented opiniondistributions of N = 100 individuals as follows: n agents were assigned to hold a random opinion drawnfrom a normal distribution ( N (0 , N − n agents were assumed to hold opinion o i = 0, thereby forming one big cluster. We varied the value of nn