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Formal models of opinion formation commonly represent an individual's opinion by a value on a fixed opinion interval. We propose an alternative modeling method wherein interpretation is only provided to the relative positions of opinions vis-\`a-vis each other. This method is then considered in a similar setting as the discrete-time Altafini model (an extension of the well-known DeGroot model), but with more general influence weights. Even in a linear framework, the model can describe, in the long run, polarization, dynamics with a periodic pattern, and (modulus) consensus formation. In addition, in our alternative approach key characteristics of the opinion dynamic can be derived from real-valued square matrices of influence weights, which immediately allows one to transfer matrix theory insights to the field of opinion formation dynamics under more relaxed conditions than in the DeGroot or discrete-time Altafini models. A few specific themes are covered: (i) We demonstrate how stable patterns in relative opinion dynamics are identified which are hidden when opinions are considered in an absolute opinion framework. (ii) For the two-agent case, we provide an exhaustive closed-form description of the relative opinion model's dynamic in the long run. (iii) We explore group dynamics analytically, in particular providing a non-trivial condition under which a subgroup's asymptotic behavior carries over to the entire population., Comment: 40 pages

Schelling and Sakoda prominently proposed computational models suggesting that strong ethnic residential segregation can be the unintended outcome of a self-reinforcing dynamic driven by choices of individuals with rather tolerant ethnic preferences. There are only few attempts to apply this view to school choice, another important arena in which ethnic segregation occurs. In the current paper, we explore with an agent-based theoretical model similar to those proposed for residential segregation, how ethnic tolerance among parents can affect the level of school segregation. More specifically, we ask whether and under which conditions school segregation could be reduced if more parents hold tolerant ethnic preferences. We move beyond earlier models of school segregation in three ways. First, we model individual school choices using a random utility discrete choice approach. Second, we vary the pattern of ethnic segregation in the residential context of school choices systematically, comparing residential maps in which segregation is unrelated to parents' level of tolerance to residential maps reflecting their ethnic preferences. Thirdly, we introduce heterogeneity in tolerance levels among parents belonging to the same group. Our simulation experiments suggest that ethnic school segregation can be a very robust phenomenon, occurring even when about half of the population prefers mixed to segregated schools. However, we also identify a sweet spot in the parameter space in which a larger proportion of tolerant parents makes the biggest difference. This is the case when the preference for nearby schools weighs heavily in parents' utility function and the residential map is only moderately segregated. Further experiments are presented that unravel the underlying mechanisms., Comment: 55 pages, 18 figures, paper presented in lecture series Institute for Advanced Studies (IAS), University of Amsterdam, March 2, 2020

Polarization between groups is a major topic of contemporary societal debate as well as of research into intergroup relations. Formal modelers of opinion dynamics try to explain how intergroup polarization can arise from simple first principles of interactions within and between groups. Models have been proposed in which intergroup attitudes affect social influence in the form of homophily or xenophobia, elaborated as fixed tendencies of individuals to interact more with in-group members, be more open to influence from in-group members and perhaps even distance oneself from attitudes of outgroup members. While these models can generate polarization between groups, their underlying assumptions curiously neglect a central insight from research on intergroup attitudes. Intergroup attitudes are themselves subject to social influence in interactions with both in- and outgroup members. I extend an existing model of opinion formation with intergroup attitudes, by adding this feedback-effect. I show how this changes model predictions about the process and the conditions of polarization between groups. In particular, it is demonstrated how the model implies that intergroup polarization can become less likely if intergroup attitudes change under social influence; and how more complex patterns of intergroup relations emerge. Especially, a renegade minority (outgroup-lovers) can have a key role in avoiding mutually negative intergroup relations and even elicit attitude reversal, resulting in a majority of individuals developing a negative attitude towards their in-group and a positive one of the outgroup. Interpretations of these theoretical results and directions for future research are further discussed., Comment: 33 pages, 21 figures, Paper presented at ODCD 2017. Interdisciplinary Workshop on Opinion Dynamics and Collective Decision 2017, July 5-7, 2017 @ Jacobs University Bremen, Germany

Empirical studies are inconclusive about the underlying mechanisms that shape the interrelated dynamics of opinions and interpersonal attraction. There is strong evidence that others whom are liked have a positive influence on opinions and similarities induce attraction (homophily). We know less about "negative" mechanisms concerning whether disliked others induce shifts away from consensus (negative influence), whether large differences (dissimilarity) generate distancing, and whether dissimilarities induce disliking (heterophobia). This study tests discriminating hypotheses about the presence of positive and negative mechanisms in controlled experiments involving dyadic interactions. Results confirm the presence of homophily, do not support the existence of negative social influence, and show a robust positive linear relationship between opinion distance and opinion shifts. This implies that contact might provide the largest push towards consensus in case of large initial differences.

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 explain clustering 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. The puzzle is particularly pressing if opinions vary continuously, such as the degree to which citizens are in favor 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 in an increasingly connected world. Considering randomness did not overcome the theoretical shortcomings so far. Small perturbations of individual opinions trigger social influence cascades that inevitably lead to monoculture, while larger noise disrupts opinion clusters and results in rampant individualism without any social structure. Our solution of the puzzle builds on recent empirical research, combining the integrative tendencies of social influence with the disintegrative effects of individualization. A key element of the new computational model is an adaptive kind of noise. We conduct simulation experiments to demonstrate that with this kind of noise, a third phase besides individualism and monoculture becomes possible, characterized by the formation of metastable clusters with diversity between and consensus within clusters. When clusters are small, individualization tendencies are too weak to prohibit a fusion of clusters. When clusters grow too large, however, individualization increases in strength, which promotes their splitting., Comment: 12 pages, 4 figures

Research on friendship networks in schools suggests that heterogeneity increases homophily preferences. We argue that this may be a misleading interpretation of the coefficients of the exponential random graph models (p*) that are used to model the network data. If students wish to avoid having no friends at all, then minority students may appear to be willing to integrate more with the majority in a more homogeneous school, even if the preference for having same ethnicity friends has the same strength in all schools. We use a random utility model of network formation to study in computational experiments the effects of the preferences that drive network choices. We generate simulated networks for different school compositions but with the same parameter for a preference for homophily. We estimate with a p* model on these simulated data the coefficients for the effect of same ethnicity on (simulated) friendship choice. The tests confirm that p* coefficients for the effect of same ethnicity are much larger in populations with a large minority, although actual homophily preferences are the same across all simulated school compositions., Comment: 20 pages, 3 figures. Paper presented at 103rd Annual Meeting of the American Sociological Association (ASA). Boston, August 4, 2008

Axelrod (1997) showed how local convergence in cultural influence can preserve cultural diversity. We argue that central implications of Axelrod's model may change profoundly, if his model is integrated with the assumption of social influence as assumed by an earlier generation of modelers. Axelrod and all follow up studies employed instead the assumption that influence is interpersonal (dyadic). We show how the combination of social influence with homophily allows solving two important problems. Our integration of social influence yields monoculture in small societies and diversity increasing in population size, consistently with empirical evidence but contrary to earlier models. The second problem was identified by Klemm et al.(2003a,b), an extremely narrow window of noise levels in which diversity with local convergence can be obtained at all. Our model with social influence generates stable diversity with local convergence across a much broader interval of noise levels than models based on interpersonal influence., Comment: 20 pages, 3 figures, Paper presented at American Sociological Association 103rd Annual Meeting, August 1-4, 2008, Boston, MA. Session on Mathematical Sociology

Recent extensions of the Axelrod model of cultural dissemination (Klemm et al 2003) showed that global diversity is extremely fragile with small amounts of cultural mutation. This seemed to undermine the original Axelrod theory that homophily preserves diversity. We show that cultural diversity is surprisingly robust if we increase the tendency towards homophily as follows. First, we raised the threshold of similarity below which influence is precluded. Second, we allowed agents to be influenced by all neighbors simultaneously, instead of only one neighbor as assumed in the orginal model. Computational experiments show how both modifications strongly increase the robustness of diversity against mutation. We also find that our extensions may reverse at least one of the main results of Axelrod. While Axelrod predicted that a larger number of cultural dimensions (features) reduces diversity, we find that more features may entail higher levels of diversity., Comment: 21 pages, 8 figures, Submitted for presentation in Mathematical Sociology Session, Annual Meeting of the American Sociological Association (ASA), 2007

We relax a simplification of Axelrod's (1997) model of cultural dissemination that has not yet been studied, the assumption that all cultural states are nominal. We integrate metric states into the original model. Computational experiments demonstrate that metric states undermine cultural diversity, even without noise, by creating sufficient overlap between agents for mutual influence. We then show how adding "bounded confidence" - a recent innovation in models of social influence - allows cultural diversity to persist. However, further experiments reveal that the solution is fragile. Diversity can be sustained only with a relatively small number of metric states, low levels of noise or narrow confidence intervals., Comment: 8 pages, 4 figures

Following Axelrod's model of cultural dissemination, formal computational studies of cultural influence have suggested that more contact between geographically distant regions may increase overall cultural homogeneity and reduce societal polarization. In the present paper, we show that two plausible modifications of Axelrod's original mechanism turn the effect of range of communication upside-down. We assume a continuous rather than a nominal state space and we add the negative side of social influence, heterophobia and rejection. Computational analyses of the resulting model demonstrate that now a larger range of contact can increase rather than decrease the extent of polarization in the population. Further experiments identify the window of conditions under which the effect obtains., Comment: 20 pages, 6 figures, Paper to be presented in the Mathematical Sociology Session of the 101st Annual Meeting of the American Sociological Association, Montreal August 11-14, 2006