1. T. Faure, G. Deffuant, G. Weisbuch, F. Amblard
Dynamics on continuous opinions probability distribution dynamics : When does extremism prevail?
T. Faure, G. Deffuant, G. Weisbuch, F. Amblard

2. Outline … Sociodynamic implementation of the Bounded Confidence model
Representation of extremists in a population
Relative Agreement (RA) Model
Sociodynamic implementation of the RA model
Convergence types (extremists win or not)
Conclusion

3. Bounded confidence model
(Deffuant et al, 2000, Krause, 2000, Hegselmann, 2001)
Continuous opinion x with an uncertainty u.
First model : all agents have the same uncertainty
Opinion dynamics : Random pair agents (xi , xj)
if :
then :
No dynamics on the uncertainty

4. With a uniform distribution of the opinions of width w
Time
[w/2u]=1 [w/2u]=2
nb attractors approximately the integer part of w/2u

5. Sociodynamic implementation of BC model
Master Equation :
Flow in
Flow out i j 1
Opinion
Simulation of the distribution evolution :

6. Opinion evolution …

7. Opinion probability density evolution …

8. Attractor number nb attractors approximately the integer part of w/2u

9. Odd Attractor number

10. Even attractor number

11. Attractor’s position
Opinion
Uncertainty

12. Population with extremists-1 +1 Ue U
Model parameters :
U : initial uncertainty of moderate agents
Ue : initial uncertainty of extremists
pe : initial proportion of extemists
d : bias between positive and negative extremists

13. New model with dynamics of uncertainties
Give more influence to more confident agents
Avoid the discontinuity of the influence when the difference of opinions grows
Explore the influence of extremists

14. Relative agreement dynamics
The modification of the opinion and the uncertainty are proportional to the relative agreement : if
 Non reciprocity for interaction
 More certain agents are more influential

15. Sociodynamic approach
Extend 1d approach : 2 D distribution of opinion and uncertainty

16. Relative agreement calculation
Opinion=0.21
Uncertainty=0.51

17. Relative agreement calculation
Opinion=0.5
Uncertainty=0.01

18. Relative agreement calculation
Opinion=0.5
Uncertainty=1

19. RA with constant uncertainty

20. RA with constant uncertainty
More attractor than BC case
1/U

21. Types of Convergence 3 types Central convergence
-Double extremes convergence
Single extreme convergence

22. Central convergence (U=0.6, pe=0.08)

23. Central convergence (U=0.6, pe=0.08)
Uncertainty
Opinion

24. Both extremes convergence (U=1., pe=0.2)

25. Single extreme convergence (U=1.4, pe=0.06)

26. Exploration of the parameter space
Description
Symbol
Tested values
Global proportion of extremists pe
0.02, 0.04, ……..,0.3
Initial uncertainty of the moderate agents U
0.2, 0.4, ….., 2
Initial uncertainty of the extremists ue
0.05, 0.1, 0.15, 0.2
Relative difference between the proportion of positive and negative extremists : d
0, 0.1, 0.2, 0.3, 0.4, 0.5
Intensity of interactions m
0.1, 0.2, 0.3, 0.4, 0.5

27. Convergence indicator
p+ and p- are the proportion of initially moderate agents which were attracted to the extreme opinion regions
y = p+2 + p-2
central convergence : y close to 0
both extreme convergence : y close to 0.5
single extreme convergence : y close to 1

28. Exploration of the parameter space

29. Conclusion
In the model, the convergence to both extremes takes place :
when the initially moderate agents are very uncertain
When the proportion of extremists is high
The convergence to a single extreme occurs when the uncertainty is even higher and the initial distribution of extremists is not exactly symmetric