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


General properties of the model

• Individual-based simulation model

• Continuous opinions

• Pair interactions

• Bounded influence


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


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


RA ModelModifications of opinion and uncertainty are

proportional to the « relative agreement »

if

(RA > 0)

More certain agents are more influential


Totally connected population


Result for u=0.5 for all


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


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


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


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


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


Unstable attractors: for the same parameters than the precedent, central

convergence


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


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


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


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)


Influence of social networks on the behaviour

of the model


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


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


First explorations on typical cases


Central convergence zonepe=0.2, U=0.4, µ=0.5, δ=0, ue = 0.1


Both extremes convergence zone pe=0.25, U=1.2, µ=0.5, δ =0, ue=0.1


Single extreme convergence zonepe=0.05, U=1.4, µ=0.5, δ = 0, ue=0.1


Basic conclusion

• Structure of the interactions / the way agents are organized influences the global behaviour of the model


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


Central convergence case (U=0.6,pe=0.05)


Both extreme convergence case

(U=1.4 pe=0.15)


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


Hypothesis

• From a connectivity value we can observe the same global phenomena than for the totally connected case


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)


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…


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)


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


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)


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


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


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


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


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


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


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


For low connectivity, it results from the aggregation of local processes of

convergence towards a single extreme


For higher connectivity, global convergence of the central cluster

which divides itself in two to converge towards each one of the

extreme


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


Thanks a lot for your attention

Some questions ?????

Date post:	18-Jan-2016
Category:	Documents
Upload:	xannon
View:	17 times
Download:	1 times

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

Documents