Discover the dynamics of bounded confidence models
We consider here one of the most popular continuous opinion models, the bounded confidence (BC) model ( ; ; ) (also proposed by ( ; ) with a different communication regime). Its principles are very close to the theory of social comparison () and to social judgment theory (). The agents are characterized by an initial opinion (generally supposed to be between -1 and 1 as in the present study, or between 0 and 1) and a threshold that can be interpreted as an uncertainty around this opinion.
The dynamics presented on this page () picks random pairs of agents that influence each other by slightly moving their opinions towards each other, but only if the distance between their opinions is lower than the uncertainty. When all the agents share the same uncertainty (on the one or on the two as presented here opinion dimensions), the opinions progressively converge to a set of opinion clusters, with all agents of a cluster sharing the same opinion and with an increasing number of clusters when the uncertainty decreases.
With variants of this model, ( ; ) study the impact of extremists on moderate agents. The extremists are initialized with extreme opinions (+1 or -1) and all share the same small uncertainty, whereas the moderates are initialized with an opinion uniformly drawn between -1 and +1 and the same larger uncertainty. By construction of the model, the extremists are more influential and less likely to change their opinions and uncertainties. Only for (), both opinions and uncertainties are modified during interactions. The extremists tend to reduce the uncertainty of the moderates during interactions. This latter model is not presented here. For , the uncertainties remain constant. For a low uncertainty of moderates or a very small part of extremists, the model is equivalent to the basic dynamic.
For the other values, depending on the initial number of extremists and the initial uncertainty of the moderates, three convergence types appear: central clusters, double extreme clusters, single extreme cluster.() exhibit a new stationary state appears when the uncertainties are fixed, for large uncertainties of the moderates. In this stationary state, the opinions of moderate agents keep fluctuating without clustering, altogether forming a stable density which shape changes significantly when the parameters vary (called here metastable state).
Instead of adding extremists in their initial population, () study dynamics including a negative influence which can also lead to the emergence of polarization and extremism. They have proposed a modelling inspired from a set of experiments (). These experiments show that highly self-engaged people in one group (ie considering it as highly self-relevant) have a different dynamics from other people: they are attracted without any conditions by members of their own group whatever the issue is and reject opinion on secondary issue if emitted by someone from another group. In the current program, we consider the dimension defining group membership in on abscissa while secondary issue are discussed on ordinate. One can notice that the presence of such agents in the population simplify the space, leading to less clusters, and can be responsible for cohesion as well polarisation.