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

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

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

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

Modelling Methodology

Modellers

Experts

Model proposalHow to improvethe model

ImplementationTheoretical study

Comparison with dataexpertise

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

Many steps and then many models…

• Cellular automata

• Agent-based models

• Threshold models

• …

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

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– …

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

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…

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

Opinion dynamics model

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

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)

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

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

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

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)

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

Homogeneous population (u=ct)

u=1.00 u=0.5

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

A brief analytical result…

• Number of clusters = [w/2u]

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

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

Heterogeneous population (ulow ,uhigh)

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

Introduction of uncertainty dynamics

• With the same condition:

• If |o-o’|<u

o=µ.(o-o’)

u=µ.(u-u’)

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

Uncertainty dynamics

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

Main problem with BC modelis the influence profile

oi

oj

oi oi+uioi-ui

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

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

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

Relative Agreement Model

Relative agreement

j

i

hij

hij-ui

oj

oi

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

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

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

– 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

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

Continuous influence

• No more sudden decrease in influence

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

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

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

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

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

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

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

Convergence cases

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

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

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

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

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

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

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

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

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

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

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

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

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

Unstable Attractors: for the same parameters than before, central

convergence

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

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

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

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

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

δ = 0, ue = 0.1, µ = 0.2, N=1000

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

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

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

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

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.

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

Tuning the balance between the two extremes

δ = 0.1, ue = 0.1, µ = 0.2

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

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

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

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)

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

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

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

First studies on network

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

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

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

Convergence casesCentral convergence

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

Both Extremes Convergence

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

Single Extreme Convergence

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

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.

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

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.)

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

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

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

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

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

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

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

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

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

Conclusion

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

computers• Submitted to the first ESSA

Conference18-22 SeptemberGröningen