Guillaume Deffuant, Frédéric Amblard, Gérard Weisbuch and Thierry Faure (2002)

How can extremism prevail? A study based on the relative agreement interaction model

Journal of Artificial Societies and Social Simulation vol. 5, no. 4
<http://jasss.soc.surrey.ac.uk/5/4/1.html>

To cite articles published in the Journal of Artificial Societies and Social Simulation, please reference the above information and include paragraph numbers if necessary

Received: 17-Jul-2002      Accepted: 20-Sep-2002      Published: 31-Oct-2002

The following set of figures, obtained from numerical situation, exhibits the three different dynamical regimes. The x-axis codes for time (number of iterations), the y-axis for opinions, and the colours for uncertainty. Each trajectory allows following the evolution in opinion and uncertainty of one individual agent.

 Common parameters are, µ = 0.5,
 ue = 0.1,
 N = 200.
 The initial uncertainty parameter u of the general population is increased from figure 5 to 7. 

u=0.4, central convergence.

Figure 5. Example of central convergence. Horizontal axis: iterations. Vertical axis: opinions. Coloured axis: uncertainties. pe = 0.2, u = 0.4, µ = 0.5, delta = 0, ue = 0.1, N = 200. The majority (96%) of the moderate agents (initially green, between the two extremes) are not attracted by the extremes.

  u = 1.2, bipolaristaion

pe = 0.25, u = 1.2, µ = 0.5, delta = 0, ue = 0.1, N = 200. The initially moderate agents (initially green, between the two extremes) split and become extremists (43% on the positive side, 56% on the negative side).

  u = 1.4, single polarisation

 shows an example of single polarisation. In this case, the majority of the population is attracted by one of the extremes. This behaviour can take place even when the number of initial extremists is the same at both extremes, which was a priori unexpected.

Single extreme convergence. The majority (98.33%) of initially moderate agents (initially green, between the two extremes) is attracted by the positive extreme.

3.11

For another sample drawn from the same initial distribution, all other parameters being equals, one can observe a central convergence (see figure 8). This dependence of which attractor is reached upon initial conditions is a signature for instability.

Figure 8. Central convergence for the same parameters as in figure 7.  The majority stays at the centre (Only one agent joins the negative extreme).

Convergence type indicator

4.1

We expressed the results of the exploration with an indicator of the convergence type, denoted y. To compute indicator y, we consider the population of opinions after convergence, and:

• we compute the proportions P+ and P-  of the initially moderate agents which became extremists in the positive extreme or negative extreme.
• The indicator is defined by:      y =   (P+)² +  (P-)²

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Typical patterns of y

4.3

Figure 9. Typical pattern of average and standard deviation of indicator y (50 simulations at each point of the graph) as a function of the uncertainty of the moderate agents (U) and the global proportion of extremists (pe) for delta = 0 (top) and delta = 0.1 (bottom). The other parameters are fixed: uncertainty of the extremists ue = 0.1, intensity of interactions µ = 0.2, initial relative difference between the extremists, delta = 0.1. On the graph of average y, the yellow or white zones on the left part correspond to central convergence, the orange, typically in the upper middle part to both extremes, and brown at the bottom right to single extreme.