Dynamique dopinions sur réseaux Amblard F.*, Deffuant G.* *C emagref-LISC.
Amblard F.* , Deffuant G.*, Weisbuch G.** *C emagref-LISC **ENS-LPS
description
Transcript of Amblard F.* , Deffuant G.*, Weisbuch G.** *C emagref-LISC **ENS-LPS
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
The drift to a single extreme appears only beyond a critical connectivity of
the social networks
Study of the relative agreement opinion dynamics on small world networks
Amblard F.*, Deffuant G.*, Weisbuch G.**
*Cemagref-LISC
**ENS-LPS
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
General properties of the model
• Individual-based simulation model
• Continuous opinions
• Pair interactions
• Bounded influence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Relative Agreement model (RA)
• N agents i – Opinion oi (uniform distrib. [–1 ; +1])
– Uncertainty ui (init. same for all)
=> Opinion segment [oi - ui ; oi + ui]
• The influence depends on the overlap between the opinion segments– No influence if they are too far– Agents are influenced in opinion and in uncertainty– The more certain, the more convincing
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
RA Model
j
i
hijui
oj
oi
Overlap : hij
Non-overlaping part : 2.ui- hij
Agreement : overlap – non-overlapAgreement : 2.(hij – ui)Relative agreement : Agreement/segmentRA : 2.(hij – ui)/2. ui = (hij – ui) / ui
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
RA ModelModifications of opinion and uncertainty are
proportional to the « relative agreement »
if
(RA > 0)
More certain agents are more influential
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Totally connected population
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Result for u=0.5 for all
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Number of clusters variation in function of u (r²=0.99)
0
2
4
6
8
10
12
0 2 4 6 8 10 12
W/2U
clus
ters
' num
ber
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Introduction of the extremists• U: initial uncertainty of the moderated agents
• ue: initial uncertainty of the extremists
• pe : initial proportion of the extremists
• δ : balance between positive and negative extremistsu
o-1 +1
U
ue
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Central convergence (pe = 0.2, U = 0.4, µ = 0.5, = 0, ue = 0.1, N = 200).
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Both extremes convergence ( pe = 0.25, U = 1.2, µ = 0.5, = 0, ue = 0.1, N = 200)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Single extreme convergence(pe = 0.1, U = 1.4, µ = 0.5, = 0, ue = 0.1, N = 200)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Unstable attractors: for the same parameters than the precedent, central
convergence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Systematic exploration
• Building of y indicator
• p’+ = prop. of moderated agents that converge to the positive extreme
• p’- = idem for the negative extreme
• y = p’+2 + p’-2
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Synthesis of the different cases for 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 of y = intermediary situations
• Variations of y in function of U and pe
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 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 convergence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Synthesis
• For a low uncertainty of the moderates (U), the influence of the extremists is limited to the nearest => central convergence
• For higher uncertainties, the extremists are more influent (bipolarisation or single extreme convergence)
• When extremists are numerous and equally distributed on the both side, instability between central convergence and single extreme convergence (due to the position of the central group + decrease of uncertainties)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Influence of social networks on the behaviour
of the model
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Adding the social network
• Before, population was totally connected, we picked up at random pairs of individuals
• Social networks: we start from a static graph, we pick up at random existing relationships (links) from this graph
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Von Neumann’s neighbourhood
• On a grid (torus)
• Each agent has got 4 neighbours (N,S,E,W)
• Advantage: more easy visualisation of the dynamics
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
First explorations on typical cases
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Central convergence zonepe=0.2, U=0.4, µ=0.5, δ=0, ue = 0.1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Both extremes convergence zone pe=0.25, U=1.2, µ=0.5, δ =0, ue=0.1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Single extreme convergence zonepe=0.05, U=1.4, µ=0.5, δ = 0, ue=0.1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Basic conclusion
• Structure of the interactions / the way agents are organized influences the global behaviour of the model
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Systematic exploration (y)
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2 0,025
0,075
0,125
0,175
0,225
0,275
Y
U
Pe
Average Y for Ue=0,1 Delta=0 Mu=0,2 N=2500 with Von Neumann Neighbourhood on a grid
0,45-0,6
0,3-0,45
0,15-0,3
0-0,15
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Central convergence case (U=0.6,pe=0.05)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Both extreme convergence case
(U=1.4 pe=0.15)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Qualitatively (VN)
• For low U : important clustering (low probability to find interlocutors in the neighbourhood, also for extremists)
• For higher U : increase of probability to find interlocutors in the neighbourhoodPropagation of the extremists’ influence until the meeting with an opposite cluster => both extreme convergence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Hypothesis
• From a connectivity value we can observe the same global phenomena than for the totally connected case
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Choice of a small-world topology
• Principle: starting from a regular structure and adding a noise for the rewiring of links
• The -model of (Watts, 1999) enables to go from regular graphs (low on the left) to random graphs (high on the right)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Change of point of view
• We choose a particular point of the space (U,pe) corresponding to a single extreme convergence (U=1.8, pe=0.05)
• We make vary the connectivity k and and try to find the single extreme convergenceagain…
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Evolution of convergence
types (y)
00,1
0,20,3
0,40,5
0,60,7
0,80,9
1
2
4
8
16
32
64
128
256
beta
k
0,9-1
0,8-0,9
0,7-0,8
0,6-0,7
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
00,1
0,20,3
0,40,5
0,60,7
0,80,9
1
2
4
8
16
32
64
128
256
beta
k
0,50-0,60
0,40-0,50
0,30-0,40
0,20-0,30
0,10-0,20
0,00-0,10in the parameter space (,k)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Remarks/Observations
• Above a connectivity of 256 (25%) we obtain the same results than the totally connected case
• When connectivity increase: Transition from both extreme convergence to single extreme convergence
• In the transition zone, high standard deviation: mix between central convergence and single extreme convergence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Explanations• Low connectivity => strong local
influence of the extremists of each side (both extremes convergence)
• For higher connectivity, higher probability to interact with the majority:– Moderates regroup at the centre– Results in a single extreme when majority
is isolated from only one of the two extremes (else central convergence)
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Explanations
• More regular is the network ( low), more the transition takes place for higher connectivity
• Regularity of the network reinforces the local propagation of extremism resulting in both extreme convergence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Influence of the network for other values of U
• Test on typical cases of convergence in the totally connected case:– Central convergence– Both extreme convergence– Single extreme convergence
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Central convergence case U=1.0
0
0,2
0,4
0,6
0,8 1 4
8
16
32
64
128
256
beta
k
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
0
0,2
0,4
0,6
0,8 1
4
16
64
256
beta
k
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Both extreme convergence case U=1.2
0
0,2
0,4
0,6
0,8 1
4
16
64
256
beta
k
0,7-0,8
0,6-0,7
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,10
0,2
0,4
0,6
0,8 1
4
16
64
256
beta
k
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Single extreme convergence case U=1.4
0
0,2
0,4
0,6
0,8 1 4
8
16
32
64
128
256
beta
k
0,9-1
0,8-0,9
0,7-0,8
0,6-0,7
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
0
0,2
0,4
0,6
0,8 1 4
16
64
256
beta
k
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Influence of the network for different values of U
• Similar dynamics• When increasing k we go from both
extreme convergence to the observed case in the totally connected case through a mix between central convergence and observed convergence in the totally connected case
• Increasing the transition takes place for lower connectivity
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Remark• In the both extreme convergence case
for the totally connected population, the two observed both extremes convergence do not correspond to the same phenomena
0
0,2
0,4
0,6
0,8 1
4
16
64
256
beta
k
0,7-0,8
0,6-0,7
0,5-0,6
0,4-0,5
0,3-0,4
0,2-0,3
0,1-0,2
0-0,1
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
For low connectivity, it results from the aggregation of local processes of
convergence towards a single extreme
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
For higher connectivity, global convergence of the central cluster
which divides itself in two to converge towards each one of the
extreme
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Perspectives
• Exploration of the influence of other parameters: µ, Ue,
• Influence of the population size (change the properties of regular graphs)
• Change of the starting structure for the small-world (2-dimension 2, generalized Moore)
• Other graphs (Scale-free networks)• Effects of the repartition of the
extremists on the graph
Frédéric Amblard – 1st ESSA Conference, Gröningen – September, 19, 2003
Thanks a lot for your attention
Some questions ?????