This paper presents and analyses a simple model of opinion dynamics when opinions can fall along continuous values. There is a bias in influence -- agents only influence each other if they are initially not that far apart. In that sense, there are similarities between this model and Axelrod's (1997) cultural dissemination model. They analyze the basic model, and then tackle extensions which introduce a network structure or a vector of (binary) opinions.
In the basic model, opinions take no continuous values. They show that if the maximum distance required for interaction is high, the whole population converges to a single value. If it is lower, then the population converges into clusters of discrete opinions, with the number of clusters inversely proportionate to the minimum distance required for interaction.
The basic model:
Consider a population of N individuals, where each individual i has an (continuously valued) opinion x_i. At each time step, some individuals are pairs. In the pair, each agent adjusts his opinion if and only if the difference of opinion is smaller in value than a threshold d, i.e., when d < |x1 – x2|. Opinions are then adjusted such that for each individual j and i in the pair,
x_i = x_i + µ*(x_j – x_i),
where µ is the convergence parameter in [0, 0.5].
Basic model results:
Simulations run with N = 1000, with initial opinions drawn from a uniform distribution in [–1, +1]. They found that the results are not dramatically changed by adjusting N and µ, and that these just influence the speed and smoothness of convergence. The big this is that lower values of d lead to more clusters.
Social Networks
They then put the agents on a lattice, such that agents could only interact with one of their 4 closest neighbors. The results were qualitatively similar, with a couple of exceptions.
1. For reasonably large d, most individuals converged, but you got pockets of isolated extremists.
2. The smaller d, a compromise position percolated through the lattice, but another compromise value was also prevalent, and you also had the persistence of extremists.
Vector Models
They then switched to binary opinions (0 or 1) and let agents have a vector of opinions of size m. An interaction occurred if paired agents afreed on at least m – d subjects. When this happens, all subjects in which they agree are conserved. When opinions on a subject differ, one (randomly selected from the pair) agent is convinced by the other with probability µ. Here are the results for m = 13:
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