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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