Models
Contributions to the literature differ in the assumptions they make about how agents select interaction partners and how they adjust their opinions as a result of the interaction.
These assumptions concern the processes internal to the agents. There is also a lot of research on the structural conditions (e.g. structure of the network) under which different macro-outcome emerge. This is summarized on the conditions site.
There are many ways to categorize models. Below, we list three important distinctions.
Nominal vs. continuous opinions
Distinction based on critical assumptions
One- vs. bi-directional influence
Nominal vs. continuous opinions
One important distinction is based on the representation of the influence dimension: While many models represent for instance opinions on nominal scales, there are also many models that focus on continuous opinions.
Nominal attributes (e.g. favorite band, political party, or movie genre). When agents hold nominal attributes, they are either maximally similar or maximally dissimilar on a given dimension. Social influence implies that agents copy the attribute of an interaction partner. A gradual influence on a given dimension is not possible.
Literature
Axelrod, R. (1997). The dissemination of culture - A model with local convergence and global polarization. Journal of Conflict Resolution, 41(2), 203–226.
Latané, B., & L’Herrou, T. (1996). Spatial Clustering in the Conformity Game: Dynamic Social Impact in Electronic Groups. Journal of Personality and Social Psychology, 70(6), 1218–1230.
Liggett, T. M. (1985). Interacting Particle Systems. New York: Springer.
Sznajd-Weron, K., & Sznajd, J. (2000). Opinion Evolution in Closed Community. International Journal of Modern Physics C,11, 1157–1165.
Continuous attributes (e.g. political orientation, left-right scale, opinion towards Donald Trump) With continuous attributes, agents can gradually change their opinions.
Literature
Abelson, R. P. (1964). Mathematical Models of the Distribution of Attitudes Under Controversy. In N. Frederiksen & H. Gulliksen (Eds.), Contributions to Mathematical Psychology(pp. 142–160). New York: Rinehart Winston.
French, J. R. P. (1956). A Formal Theory of Social Power. Psychological Review, 63(3), 181–194.
Friedkin, N. E., & Johnsen, E. C. (2011). Social Influence Network Theory. New York: Cambridge University Press.
Macy, M. W., Kitts, J., Flache, A., & Benard, S. (2003). Polarization and Dynamic Networks. A Hopfield Model of Emergent Structure. In R. Breiger, K. Carley, & P. Pattison (Eds.), Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers(pp. 162–173). Washington, DC: The National Academies Press.
Mäs, M., Flache, A., & Helbing, D. (2010). Individualization as driving force of clustering phenomena in humans. PLoS Computational Biology, 6(10).
Distinction based on critical assumptions
A second very important classification categorizes models according to the assumptions they make about how agents influence each other. We focus here on models assuming opinion measured on a continuous scale, as this continuous scales allow more diverse assumptions. However, social influence can also be implemented in different ways when opinions are nominal. For instance, some models assume that agents copy a trait from an interaction partner, while alternative models assume that agents adopt the most frequent trait in their network.
Positive social influence
Classical models of social influence assume that actors grow always more similar when they influence each other. In models studing continuous opinions, this is often implemented as averaging. The most important prediction of these models is that perfect consensus is unevitable unless the network is segregated into unconnected components.
Literature
Abelson, R. P. (1964). Mathematical Models of the Distribution of Attitudes Under Controversy. In N. Frederiksen & H. Gulliksen (Eds.), Contributions to Mathematical Psychology(pp. 142–160). New York: Rinehart Winston.
French, J. R. P. (1956). A Formal Theory of Social Power. Psychological Review, 63(3), 181–194.
Moderated positive influence and bounded confidence
In these models, influence is also implemented as positive social influence, but it is added that influence is limited when agents disagree. Bounded-confidence models, in fact, add that there is zero influence when differences exceed a given threshold. The bounded-confidence assumption makes it possible to explain the emergence of opinion fragmentation. If no further assumptions are added, however, these models fail to explain the emergence of opinion polarization.
Literature
Carley, K. (1991). A Theory of Group Stability. American Sociological Review, 56(3), 331–354.
Deffuant, G., Huet, S., & Amblard, F. (2005). An Individual-Based Model of Innovation Diffusion Mixing Social Value and Individual Benefit. American Journal of Sociology, 110(4), 1041–1069.
Hegselmann, R., & Krause, U. (2002). Opinion Dynamics and Bounded Confidence Models, Analysis, and Simulation. Journal of Artificial Societies and Social Simulation, 5(3).
Kurahashi-Nakamura, T., Mäs, M., & Lorenz, J. (2016). Robust clustering in generalized bounded confidence models. Journal of Artificial Societies and Social Simulation,19(4).
Lorenz, J. (2007). Continuous opinion dynamics under bounded confidence: A survey. International Journal of Modern Physics C,18(12), 1819–1838.
Impact of the initial opinion / Stubbornness
According to the famous Friedkin-Johnson model, influence is modeled as averaging. Influence, however, is limited in that agents are stubborn and tend to be "pulled back" by their initial opinion. This can explain why positive influence does not always lead to the emergence of consensus.
Literature
Friedkin, N. E., & Johnsen, E. C. (2011). Social Influence Network Theory. New York: Cambridge University Press.
Negative social influence
Negative-influence models assume that agents grow more similar when they like each other. If they dislike each other, however, they tend to intensify differences. This assumption is often combined with `heterophilia', the tendency to dislike indiviuals who disagree. Negative influence is a critical assumption, as it is one of the few mechanisms that can explain the emergence of opinion polarization.
Literature
Flache, A., & Mäs, M. (2008). How to get the timing right. A computational model of the effects of the timing of contacts on team cohesion in demographically diverse teams. Computational and Mathematical Organization Theory, 14(1), 23–51.
Macy, M. W., Kitts, J., Flache, A., & Benard, S. (2003). Polarization and Dynamic Networks. A Hopfield Model of Emergent Structure. In R. Breiger, K. Carley, & P. Pattison (Eds.), Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers(pp. 162–173). Washington, DC: The National Academies Press.
Salzarulo, L. (2006). A Continuous Opinion Dynamics Model Based on the Principle of Meta-Contrast. Journal of Artificial Societies and Social Simulation, 9(1).
Striving for uniqueness
These models also assume negative influence, but agents are assumed to distance themselves from similar others, for instance because they seek to be unique. These models are hardly studied, but they imply very rich opinion dynamics.
Literature
Mäs, M., Flache, A., & Helbing, D. (2010). Individualization as driving force of clustering phenomena in humans. PLoS Computational Biology, 6(10).
Mäs, M., Flache, A., & Kitts, J. A. (2014). Cultural Integration and Differentiation in Groups and Organizations. In V. Dignum & F. Dignum (Eds.), Perspectives on Culture and Agent-based Simulations. Cham: Springer International Publishing.
Smaldino, P. E., & Epstein, J. M. (2015). Social conformity despite individual preferences for distinctiveness. Royal Society Open Science, 2(3).
Noise
Many models make markedly different predictions when their assumptions are implemented probabilistically rather than deterministically. Noise can foster both consensus and opinion differences, dependening on how it is implemented.
Literature
Kurahashi-Nakamura, T., Mäs, M., & Lorenz, J. (2016). Robust clustering in generalized bounded confidence models. Journal of Artificial Societies and Social Simulation, 19(4).
Mäs, M., Flache, A., & Helbing, D. (2010). Individualization as driving force of clustering phenomena in humans. PLoS Computational Biology, 6(10).
Pineda, M., Toral, R., & Hernandez-Garcıa, E. (2009). Noisy continuous-opinion dynamics. Journal of Statistical Mechanics,P08001.
Opinion reinforcement
Opinion-reinforcement models assume that agents grow more extreme when they interact with someone who holds similar views. One mechanism underlying opinion reinforcement is the communication of persuasive arguments: opinions are reinforced because agents provide each other with further reasons supporting their opinions. Argument communication leads to opinion polarization when actors with similar opinions interact and, therefore, reinforce each other's opinions. These models are able to explain the emergence of opinion polarization without assuming negative influence.
Literature
Dandekar, P., Goel, A., & Lee, D. T. (2013). Biased assimilation, homophily, and the dynamics of polarization. Proceedings of the National Academy of Sciences of the United States of America, 110(15), 5791–6.
Mäs, M., & Flache, A. (2013). Differentiation without distancing. Explaining bi-polarization of opinions without negative influence. PLoS ONE, 8(11).
Mäs, M., Flache, A., Takács, K., & Jehn, K. (2013). In the short term we divide, in the long term we unite: Demographic crisscrossing and the effects of faultlines on subgroup polarization. Organization Science, 24(3), 716–736.
Banisch, S., & Olbrich, E. (2019). Opinion polarization by learning from social feedback. The Journal of Mathematical Sociology, 43(2), 76-103.
Public vs. private opinions
Another mechanism that contributes to explaining the emergence of opinion polarization assumes that agents do not communicate their private opinions but publicly adopt opinions on the poles of the opinions. Thus, others are influenced not by the private opinions but only the public ones. Similar models follow when agents have private beliefs but obseverve each other's behavior.
Literature
Stubborn extremists
Modelers sometimes add to the bounded-confidence model (see above) that actors with extreme opinions are more less open to influence. This additional assumption changes model predictions very much. In particular, it can generate opinion polarization.
One- vs. bi-directional influence
A third distinction focusses on whether influence is one- or bi-directional. Sometimes influence is one-directional in that an actor A can influence B but not the other way around. For instance, viruses can be transmitted from an infected person to a healthy person, but a healthy person cannot heal an infected person. This form of social influence is models with diffusion models.
The models that we collect on this website, on the other hand, asssume that influence is bi-directional. That is, agents can influence each other independent if their current state. A conservative can exert social influence on a liberal and the other way around.