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

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 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

RA Model Modifications 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)

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

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).

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)

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)

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 zone pe=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 zone pe=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)

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 neighbourhood Propagation 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 convergence again…

Evolution of convergence types (y) in 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

Both extreme convergence case U=1.2

Single extreme convergence case U=1.4

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

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 ?????