1. Amblard F.*, Deffuant G.*, Weisbuch G.** *Cemagref-LISC **ENS-LPS
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

2. General properties of the model
Individual-based simulation model
Continuous opinions
Pair interactions
Bounded influence

3. 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

4. RA Model Overlap : hij Non-overlaping part : 2.ui- hij
Agreement : overlap – non-overlap
Agreement : 2.(hij – ui)
Relative agreement : Agreement/segment
RA : 2.(hij – ui)/2. ui = (hij – ui) / ui j i
hij ui oj oi

5. RA Model
Modifications of opinion and uncertainty are proportional to the « relative agreement » if
(RA > 0)
 More certain agents are more influential

6. Totally connected population

7. Result for u=0.5 for all

8. Number of clusters variation in function of u (r²=0.99)

9. 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 extremists u o -1 +1 U ue

10. Central convergence (pe = 0. 2, U = 0. 4, µ = 0. 5,  = 0, ue = 0
Central convergence (pe = 0.2, U = 0.4, µ = 0.5,  = 0, ue = 0.1, N = 200).

11. Both extremes convergence ( pe = 0. 25, U = 1. 2, µ = 0
Both extremes convergence ( pe = 0.25, U = 1.2, µ = 0.5,  = 0, ue = 0.1, N = 200)

12. Single extreme convergence (pe = 0. 1, U = 1. 4, µ = 0
Single extreme convergence (pe = 0.1, U = 1.4, µ = 0.5,  = 0, ue = 0.1, N = 200)

13. Unstable attractors: for the same parameters than the precedent, central convergence

14. 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

15. 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

16. 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

17. 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)

18. Influence of social networks on the behaviour of the model

19. 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

20. 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

21. First explorations on typical cases

22. Central convergence zone pe=0.2, U=0.4, µ=0.5, δ=0, ue = 0.1

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

24. Single extreme convergence zone pe=0.05, U=1.4, µ=0.5, δ = 0, ue=0.1

25. Basic conclusion
Structure of the interactions / the way agents are organized influences the global behaviour of the model

26. Systematic exploration (y)

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

28. Both extreme convergence case (U=1.4 pe=0.15)

29. 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 neighbourhood
Propagation of the extremists’ influence until the meeting with an opposite cluster => both extreme convergence

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

31. 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)

32. 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
convergence
again…

33. Evolution of convergence types (y)
in the parameter space (,k)

34. 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

35. 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)

36. 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

37. 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

38. Central convergence case U=1.0

39. Both extreme convergence case U=1.2

40. Single extreme convergence case U=1.4

41. 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

42. 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

43. For low connectivity, it results from the aggregation of local processes of convergence towards a single extreme

44. For higher connectivity, global convergence of the central cluster which divides itself in two to converge towards each one of the extreme

45. 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

46. Thanks a lot for your attention
Some questions ?????