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

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

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

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

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

Opinion evolution …

Opinion probability density evolution …

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

Odd Attractor number

Even attractor number

Attractor’s position Opinion Uncertainty

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

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

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

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

Relative agreement calculation Opinion=0.21 Uncertainty=0.51

Relative agreement calculation Opinion=0.5 Uncertainty=0.01

Relative agreement calculation Opinion=0.5 Uncertainty=1

RA with constant uncertainty

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

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

Central convergence (U=0.6, pe=0.08)

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

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

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

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

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

Exploration of the parameter space

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