Amblard F. , Deffuant G., Weisbuch G. C emagref-LISC ENS-LPS

Frédéric Amblard - RUG-ICS Meeting - June 12, 2003

How can extremism prevail?

An opinion dynamics model studied with heterogeneous agents and networks

Amblard F., Deffuant G., Weisbuch G.

Cemagref-LISC

ENS-LPS


Context

• European project FAIR-IMAGES• Modelling the socio-cognitive processes of

adoption of AEMs by farmers• 3 countries (Italy, UK, France)• Interdisciplinary project

– Economics– Rural sociology– Agronomy– Physics– Computer and Cognitive Sciences


Modelling Methodology

Modellers

Experts

Model proposalHow to improvethe model

ImplementationTheoretical study

Comparison with dataexpertise


Many steps and then many models…

• Cellular automata

• Agent-based models

• Threshold models

• …


Final (???) model…

• Huge model integrating:– Multi-criteria decision (homo socio-economicus)– Expert systems (economic evaluation)– Opinion dynamics model– Information diffusion– Institutional action (scenarios)– Social networks– Generation of virtual populations– …


Using/understanding of the final model

• Using the model as a data transformation (inputs->model->outputs) we study correlations between inputs and outputs…

• Model highly stochastic, then many replications

• To understand the correlations?– We have to get back to basics… – Study each one of the component independently…


Opinion dynamics model


Bibliography

• Opinion dynamics models– Models of binary opinions and vote models

(Stokman and Van Oosten, Latané and Nowak, Galam, Galam and Wonczak, Kacpersky and Holyst)

– Models with continuous opinions, negotiation framework, collective decision-making (Chatterjee and Seneta, Cohen et al., Friedkin and Johnsen)

– Threshold Models (BC) (Krause, Deffuant et al., Dittmer, Hegselmann and Krause)


Opinion dynamics model

• Basic features:– Agent-based simulation model– Including uncertainty about current opinion– Pair interactions– The less uncertain, the more convincing– Influence only if opinions are close enough– When influence, opinions move towards

each other


First model (BC)• Bounded Confidence Model• Agent-based model• Each agent:

– Opinion o [-1;1] (Initial Uniform Distribution)

– Uncertainty u +

– Pair interaction between agents (a, a’)– If |o-o’|<u

o=µ.(o-o’)– µ = speed of opinion change = ct– Same dynamics for o’– No dynamics on uncertainty (at this stage)


Homogeneous population (u=ct)

u=1.00 u=0.5


A brief analytical result…

• Number of clusters = [w/2u]

– w is width of the initial distribution– u the uncertainty


Heterogeneous population (ulow ,uhigh)


Introduction of uncertainty dynamics

• With the same condition:

• If |o-o’|<u

o=µ.(o-o’)

u=µ.(u-u’)


Uncertainty dynamics


Main problem with BC modelis the influence profile

oi

oj

oi oi+uioi-ui


Relative Agreement Model (RA)

• N agents i – Opinion oi (init. uniform distrib. [–1 ; +1])– Uncertainty ui (init. ct. for the population)– Opinion segment [oi - ui ; oi + ui]

• Pair interactions• Influence depends on the overlap between opinion

segments– No influence if they are too far– The more certain the more convincing– Agents are influenced each other in opinion and

uncertainty


Relative Agreement Model

Relative agreement

j

i

hij

hij-ui

oj

oi


Relative Agreement Model

Modifications of the opinion and the uncertainty are proportional to the “relative agreement”

hij is the overlap between the two segments

if

Most certain agents are more influential


– Continuous interaction functions

o o-u o+u

o’+u’o’o’-u’

h 1-h

o o-u o+u

o’+u’o’o’-u’

h 1-h


Continuous influence

• No more sudden decrease in influence


Result with initial u=0.5 for all

0

2

4

6

8

10

12

0 2 4 6 8 10 12

W/2U

clus

ters

' num

ber


Constant uncertainty in the population u=0.3

(opinion segments)

0

10

20

30

40

50

60

-1,3

-1,1

-0,8

-0,6

-0,4

-0,1 0,1 0,4 0,6 0,8 1,1 1,3

0

100

200

nb

t

opinions


Introduction of extremists• U : initial uncertainty of moderated agents

• ue : initial uncertainty of extremists

• pe : initial proportion of extremists

• δ : balance between positive and negative extremistsu

o-1 +1


Convergence cases


Central convergence (pe = 0.2, U = 0.4, µ = 0.5, = 0, ue = 0.1, N = 200).


Central convergence(opinion segments)

0

24

48

72

96

120

-1,1 -0,8 -0,6 -0,3 -0,1 0,2 0,5 0,7 1,0 1,2

0

50

100

150

200

nb

t

opinions


Both extremes convergence ( pe = 0.25, U = 1.2, µ = 0.5, = 0, ue = 0.1, N = 200)


Both extremes convergence(opinion segment)

0

24

48

72

96

120

-1,1 -0,8 -0,6 -0,3 -0,1 0,2 0,5 0,7 1,0 1,2

0

50

100

150

200

250

nb

t

opinions


Single extreme convergence(pe = 0.1, U = 1.4, µ = 0.5, = 0, ue = 0.1, N = 200)


Single extreme convergence(opinion segment)

0

40

80

120

160

200

240

-1,1

-0,9 -0,7 -0,5

-0,3 -0,1 0,1 0,3 0,5 0,7 0,9 1,1

0100200300400

nb

t

opinions


Unstable Attractors: for the same parameters than before, central

convergence


Systematic exploration

• Introduction of the indicator y

• p’+ = prop. of moderated agents that converge to positive extreme

• p’- = prop. Of moderated agents that converge to negative extreme

• y = p’+2 + p’-2


Synthesis of the different cases with y

• Central convergence– y = p’+2

+ p’-2 = 0² + 0² = 0

• Both extreme convergence– y = p’+2

+ p’-2 = 0.5² + 0.5² = 0.5

• Single extreme convergence– y = p’+2

+ p’-2 = 1² + 0² = 1

• Intermediary values for y = intermediary situations

• Variations of y in function of U and pe


δ = 0, ue = 0.1, µ = 0.2, N=1000

(repl.=50)• white, light yellow => central convergence• orange => both extreme convergence• brown => single extreme


What happens for intermediary zones?

• Hypotheses:– Bimodal distribution of pure attractors (the

bimodality is due to initialisation and to random pairing)

– Unimodal distribution of more complex attractors with different number of agents in each cluster


pe = 0.125 δ = 0

(U > 1) => central conv. Or single extreme (0.5 < U < 1) => both extreme conv. (u < 0.5) => several convergences between central and both extreme conv.


Tuning the balance between the two extremes

δ = 0.1, ue = 0.1, µ = 0.2


Influence of the balance(δ = 0;0.1;0.5)


Conclusion• For a low uncertainty of the moderate (U), the

influence of the extremists is limited to the nearest => central convergence

• For higher uncertainties in the population, extremists tend to win (bipolarisation or conv. To a single extreme)

• When extremists are numerous and equally distributed on the both sides, instability between central convergence and single extreme convergence (due to the position of the central group + and to the decrease of the uncertainties)


Modèle réalisé• Modèle stochastique• Trois types de liens :

– Voisinage– Professionnels– Aléatoires

• Attribut des liens :– Fréquence d’interactions

• Paramètres du modèles :– densité et fréquence de chacun des types, – dl, relation d’équivalence pour les liens

professionnels


First studies on network


Network topologies• At the beginning:

– Grid (Von Neumann and De Moore neighbourhoods) => better visualisation

• What is planned– Small World networks (especially β-model

enabling to go from regular networks to totally random ones)

– Scale-free networks

• Why focus on “abstract” networks?– Searching for typical behaviours of the

model– No data available


Convergence casesCentral convergence


Both Extremes Convergence


Single Extreme Convergence


Schematic behaviours

• Convergence of the majority towards the centre

• Isolation of the extremists (if totally isolated => central convergence)

• If extremists are not totally isolated– If balance between non-isolated

extremists of both side => double extr. conv.

– Else => single extr. conv.


Problems

• Criterions taken for the totally connected case does not enable to discriminate

• With networks => more noisy situation to analyse…

• Totally connected case => only pe, delta and U really matters

• Network case– Population size– Ue matters (high Ue valorise central conv.)


Nb of iteration to convergence

0,2

0,5

0,8

1,1

1,4

1,7

2,0

0,025

0,1

0,175

0,25

nb moyen d interactions

U

Pe

temps de convergence moyen connectivité=4 delta=0

450000,00-500000,00

400000,00-450000,00

350000,00-400000,00

300000,00-350000,00

250000,00-300000,00

200000,00-250000,00

150000,00-200000,00

100000,00-150000,00

50000,00-100000,00

0,00-50000,00


Nb of clusters (VN)0,

2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,025

0,075

0,125

0,175

0,225

0,275

U

Pe

Nb Clusters connectivité=4 delta=0

500,00-600,00

400,00-500,00

300,00-400,00

200,00-300,00

100,00-200,00

0,00-100,00


Nb clusters (dM)0,

2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,025

0,075

0,125

0,175

0,225

0,275

U

Pe

Nb Clusters connectivité=8 delta=0

300,00-400,00

200,00-300,00

100,00-200,00

0,00-100,00


Network efficience0,

2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,025

0,075

0,125

0,175

0,225

0,275

U

Pe

Efficience du réseau connectivité=4 delta=0

0,90-1,00

0,80-0,90

0,70-0,80

0,60-0,70

0,50-0,60

0,40-0,50

0,30-0,40

0,20-0,30


Conclusion

• Many simulations to do…• Currently running on a cluster of

computers• Submitted to the first ESSA

Conference18-22 SeptemberGröningen

Documents

Amblard F. , Deffuant G., Weisbuch G. C emagref-LISC ENS-LPS