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Mathematical models for opinion dynamics
Laurent Boudin
Lab. Jacques-Louis Lions, Univ. Pierre et Marie Curie & Reo, Inria
PDE models in social sciences – January 24th, 2014Closure of Peter Markowich’s chair – FSMP
Joint works withAurore Mercier, Roberto Monaco and Francesco Salvarani
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 1 / 44
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Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 2 / 44
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Sociophysics
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 3 / 44
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Sociophysics
What is sociophysics?
Using methods and concepts from Physicsto describe political and social behaviours.
Notion popularized by Galam during the 80s.
Opinion formation, minorities’ power, rumours,political alliances, media influence, etc.
But what is its validity?
Galam predicted the result of the referendum on the Europeanconstitution in France several months before the votes.
He also predicted the scenario of the 2002 French presidential election.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 4 / 44
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Sociophysics
Complex systems modelling
Ising models: mainly developed by Galam et al., starting from thepioneering work by Galam, Gefen and Shapir in 1982.
Competition between various deterministic phenomena in a groupmodelled as a multi-agent system:
I Weisbuch, Deffuant...I Weidlich, Hegselmann, Krause...
Collective behaviour of a large number of individuals=⇒ possible kinetic approaches.
Kinetic models for social dynamics: Helbing.
Opinion formation in a closed community since 2000:I Ben-Naim...I Toscani; LB, Salvarani; Düring, Markowich, Pietschmann, Wolfram...
Main feature of all models: tendency to compromise.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 5 / 44
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Compromise vs. self-thinking
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 6 / 44
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Compromise vs. self-thinking Model
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 7 / 44
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Compromise vs. self-thinking Model
Modelling
Opinions regarding a binary question (like referendum)
Opinion variable x ∈ [−1, 1] = Ω̄x = ±1 ⇐⇒ “yes” / “no” without reservex = 0 ⇐⇒ no preference
Fraction (number) f (t, x) dx of individuals with opinion x at time tNatural functional framework: f (t, ·) ∈ L1(Ω)
Competition between two processes:I Interaction between two individuals (collisional process)I Self-thinking (diffusive process)
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 8 / 44
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Compromise vs. self-thinking Model
Diffusion operator
Self-thinking (Ben-Naim 2005) changes opinion: governed by ∂x(α∂x f ).
Definition
Diffusion function α ∈ C 1(Ω̄,R) admissible if α is even and α(±1) = 0.
Example
α(x) = κ(1− x2)r , r , κ > 0:individuals with a strong opinion are more stable in their conviction.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 9 / 44
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Compromise vs. self-thinking Model
Collision mechanismStronger opinions less attracted to the average than weaker opinions
x ′ =x + x∗
2+ η(x)
x − x∗2
x ′∗ =x + x∗
2− η(x∗)
x − x∗2
Definition
Attraction function η ∈ C 1(Ω̄,R) admissible ifη is even and 0 ≤ η < 1,η′ > 0 for x > 0,
Jacobian J of the collision rule s.t. J(x , x∗) ≥ J0 > 0.
Example
η(x) = λ(1 + x2), 0 < λ < 1/2, is admissible (J ≥ λ).
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 10 / 44
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Compromise vs. self-thinking Model
Collision operator Q of Boltzmann type
The collision mechanism is not a diffeomorphism of Ω̄2
=⇒ use of a weak form:
〈Q(f , f ), φ〉 = β∫∫
Ω2f (t, x)f (t, x∗)
[φ(x ′)− φ(x)
]dx∗ dx
Parameter β is the constant collision frequency.
Gain term Q+(f , f ): quantifies the exchanges of opinions betweenindividuals which give a post-collisional opinion x .
Loss term Q−(f , f ): quantifies the exchanges of opinions where anindividual with pre-collisional opinion x interacts with another one.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 11 / 44
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Compromise vs. self-thinking Model
A priori estimates on Q+ and Q
Lemma
‖Q+(f , f )(t, ·)‖L1(Ω) ≤2β
1−max η‖f (t, ·)‖2L1(Ω)
‖Q(f , f )(t, ·)‖L1(Ω) ≤(
2
1−max η+ 1
)β ‖f (t, ·)‖2L1(Ω)
We just have to write the gain term under the form
〈Q+(f , f ), φ〉 = β∫∫
Ω2Kη(x , x
′) f (t, ξ(x , x ′)) f (t, x ′) φ(x) dx ′ dx .
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 12 / 44
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Compromise vs. self-thinking Model
Weak form of the model〈∂t f − ∂x(α∂x f )− Q(f , f ), ϕ
〉= 0 a.e. t ∈ [0,T ]
Initial datum: f in ≥ 0 in Ω̄ Test functions: ϕ ∈ C 2(Ω̄)
Proposition
Mass conservation: ‖f (t, ·)‖L1(Ω) = ‖f in‖L1(Ω)
Bounded moments:
∫Ω|x |nf (t, x) dx ≤ ‖f in‖L1(Ω)
Mass conservation is not realistic for long-time forecasts. In the case of areferendum, we only study short-time forecasts.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 13 / 44
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Compromise vs. self-thinking Existence result and proof
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 14 / 44
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Compromise vs. self-thinking Existence result and proof
Existence result
Theorem
There exists a nonnegative weak solution f ∈ L∞t (L1x) to our problem, forany initial datum f in ≥ 0.
Sketch of the proof
1 Build a sequence of approximate problems.
2 Obtain the existence of solutions (f n)n∈N to such problems.
3 Study the convergence of that sequence.
4 Check the limit solves our problem.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 15 / 44
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Compromise vs. self-thinking Existence result and proof
Approximate problems
Set % =
∫Ωf in(x∗) dx∗.
Build (f n)n∈N
Consider (f n) defined by f 0 ≡ 0 and (in the distributional sense)
∂t fn+1 − ∂x(α∂x f n+1) + β%f n+1 = Q+(f n, f n)
with initial condition f n(0, ·) = f in and boundary conditions
limx→±1
α(x)∂x fn(t, x) = 0.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 16 / 44
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Compromise vs. self-thinking Existence result and proof
Preliminary result
Proposition
Consider the IBVP in Ω× [0,T ]
∂tv − ∂x(α∂xv) + µv = g , µ ≥ 0, g ∈ C ([0,T ]; L1(Ω)), g ≥ 0
with initial and boundary conditions
v(0, ·) = v in ∈ L1(Ω), v in ≥ 0, limx→±1
α(x)∂xv(t, x) = 0
There exists a unique solution v ∈ C 0([0,T ]; L1(Ω)), and v ≥ 0.
If µ = 0 and g = 0, see Campiti, Metafune, Pallara, 1998.Conclude for other cases thanks to the Duhamel principle.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 17 / 44
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Compromise vs. self-thinking Existence result and proof
Study of the solutions to the approximate problems
Using the preliminary result and the a priori estimates on Q+, we candeduce the existence of f n ≥ 0 in C 0([0,T ]; L1(Ω)).Choosing ϕ ≡ 1 as a test-function implies
d
dt
∫Ωf n+1 dx + β%
∫Ωf n+1 dx = β
(∫Ωf n dx
)2.
By induction,
∫Ωf n dx ≤ %.
Besides, (f n) is a non-decreasing sequence(study the eqn. satisfied by f n+1 − f n).By monotone convergence, (f n) converges to f ∈ L∞t (L1x) and a.e.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 18 / 44
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Compromise vs. self-thinking Existence result and proof
Eventually...Let us prove that f solves our problem in its weak form:∫ T
0
∫Ωf n+1 ϕ(x) ψ(t) dx dt −
∫ T0
∫Ω
(α(x) ϕ′(x)
)′f n+1 ψ(t) dx dt
+ β%
∫ T0
∫Ωf n+1 ϕ(x) ψ(t) dx dt
= β
∫ T0
∫∫Ω2
f n(t, x)f n(t, x∗)ϕ(x′)ψ(t) dx dx∗ dt.
Mass conservation implies convergence OK for the third integral.
The fourth integral also converges since∫∫Ω2|f n(x)f n(x∗)− f (x)f (x∗)| |ϕ(x ′)| dx dx∗
≤ 2% ‖f n(t, ·)− f (t, ·)‖L1 ‖ϕ‖L∞ .
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 19 / 44
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Compromise vs. self-thinking Numerical experiments
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 20 / 44
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Compromise vs. self-thinking Numerical experiments
Initial uniformly distributed opinions with no diffusion
f in ≡ 1/2, κ = 0
Æ" #! #Æ" #! " # $ $ ! $! �! "!!�#!& " $%%%
0
0.2
0.4
0.6
0.8
1
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
opinions
" %'& $ " %'& & " #%%%Asymptotic formation of one peak centred on the average opinion (0 inour case, 0.5 in Deffuant et al.)
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 21 / 44
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Compromise vs. self-thinking Numerical experiments
Initial uniformly distributed opinions with diffusion
f in ≡ 1/2, κ = 0.05
Very fast asymptotic equilibrium of f , I+ =
∫ 10
f (t, x) dx lies between
48% and 52%.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 22 / 44
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Compromise vs. self-thinking Numerical experiments
Two balanced opposite biases case
Centred bimodal initial datum, κ = 0.05Similar experiment in Galam, 1997
Very fast asymptotic equilibrium of f , I+ varies between 48% and 52%.Conclusion: within a balanced representation group, exchange of opinionsfavours compromise.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 23 / 44
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Compromise vs. self-thinking Numerical experiments
Two unbalanced opposite biases case
f in(x) = 1 if −1 ≤ x ≤ −1/2, Dirac mass in +1, 0 elsewhere, κ = 0.05Similar experiment in Galam, 1997
Very fast asymptotic equilibrium of f , I+ almost goes to 100%.Conclusion: within an unbalanced representation group, exchange ofopinions favours the initially strongest representation.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 24 / 44
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Contradictory agents
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 25 / 44
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Contradictory agents
Conciliatory vs. contradictory people
Main goal of this work: breaking the long-time equilibrium.
Two groups:I conciliatory individuals: distribution function f ,I contradictory people: distribution function g .
Assumption (probably questionable):mass conservation of each group.
Coupled homogeneous kinetic equations of Boltzmann type
∂t f = Q(f , f ) + R1(f , g)
∂tg= R2(f , g) + S(g , g)
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 26 / 44
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Contradictory agents
Collision rules for the interaction
xR=x + x∗
2+ η(x)
x − x∗2
xR∗ =
1− (1− xR)(1− x∗)
(1− x)if x < x∗
x∗ if x = x∗
(1 + xR)(1 + x∗)
(1 + x)− 1 if x > x∗
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 27 / 44
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Contradictory agents
Disjoint-supported initial data (1)
f in(x) =
{2 ifx < −0.50 ifx > −0.5 g
in(x) =
{0 ifx < 0.50.2 ifx > 0.5
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 28 / 44
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Contradictory agents
Disjoint-supported initial data (2)
I+(f + g) I+(f )
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 29 / 44
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Media influence, multipartite system
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 30 / 44
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Media influence, multipartite system Multidimensional model
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 31 / 44
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Media influence, multipartite system Multidimensional model
Modelling
Competition between two processes:I Microscopic interaction between two individuals (nonlinear)I Interaction with a time-dependent background (linear)
p political parties, m mass media
Opinion vector x ∈ [−1, 1]p = Ω̄pxi = ±1 ⇐⇒ pro/anti party i without reservexi = 0 ⇐⇒ no preference w.r.t. party i
Percentage (number) f (t, x) dx of individuals with opinion x at time tNatural functional framework: f (t, ·) ∈ L1(Ωp)
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 32 / 44
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Media influence, multipartite system Multidimensional model
Collision operator
Binary interactions: for any 1 ≤ i ≤ p,x ′i =
xi + x∗i
2+ η(xi )
xi − x∗i2
(x∗i )′ =
xi + x∗i
2− η(x∗i )
xi − x∗i2
The attraction function η can depend on i .
Weak form
〈Q(f , f ), φ〉 = β∫∫
Ω2pf (t, x)f (t, x∗)
[φ(x ′)− φ(x)
]dx∗ dx
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 33 / 44
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Media influence, multipartite system Multidimensional model
Linear operator
Media characteristics, for any 1 ≤ j ≤ mI The strength αj measures the media influence
w.r.t. another one and the binary interactions.I The media opinion vector X j ∈ Ω̄p
states the agreement of media j w.r.t. each party, can depend on time.
Microscopic interaction with media, for any i , j
x̃i = xi + ξj(|X ji − xi |)(Xji − xi )
ξj ∈ C 1([0, 2]; [0, 1)) influence function, decreasing, and nil beyond agiven threshold, e.g. ξj(s) = (1 + cos(πs))/4 on [0, 1], 0 elsewhere
Weak form
〈Lj f , φ〉 = αj∫
Ωpf (t, x) [φ(x̃)− φ(x)] dx
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 34 / 44
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Media influence, multipartite system Mathematical result
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 35 / 44
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Media influence, multipartite system Mathematical result
Mathematical result
Full model
∂t f =m∑j=1
Lj f + Q(f , f ) in [0,T ]× Ω
Test functions: ϕ ∈ C 0(Ωp)The total mass is still conserved.
Theorem
There exists a unique nonnegative weak solution f ∈ C 0t (L1x) to theproblem, for any L1 initial datum f in ≥ 0.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 36 / 44
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Media influence, multipartite system Two-party numerical experiments
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 37 / 44
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Media influence, multipartite system Two-party numerical experiments
Influence of one unique media 1/2
Media opinion: (0.9, 0.9), strength: α1 = 0.1β
Situation 1 – Concentration effect.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 38 / 44
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Media influence, multipartite system Two-party numerical experiments
Influence of one unique media 2/2
Media opinion: (0.9, 0.9), strength: α1 = 0.1β
Situation 2 – Threshold effect.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 39 / 44
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Media influence, multipartite system Two-party numerical experiments
Opinion manipulation by a mediaMedia opinions: (0.4,−0.4) and (−0.4, 0.4)/(−0.39, 0.39),
strength: α1 = α2 = 0.1β
I1 =
∫∫x1>x2
f (x) dx
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 40 / 44
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Media influence, multipartite system Three-party numerical experiment
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 41 / 44
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Media influence, multipartite system Three-party numerical experiment
Strong media with variable opinionMedia opinions: (−0.2, 0.3,−0.2),(−0.2,−0.2, 0.3)
(0.3,−0.2,−0.2) + 0.2 cos(2πt/100)(−1, 1, 1),strength: α3 = 0.2β, α1 = α2 = 0.1β
I1 =
∫∫x1>max(x2,x3)
f (x) dx , I3
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 42 / 44
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Conclusion and prospects
Outline
1 Sociophysics
2 Compromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
3 Contradictory agents (with A. Mercier, F. Salvarani)
4 Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
5 Conclusion and prospects
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 43 / 44
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Conclusion and prospects
Conclusion and prospects
Numerous kinetic models for opinion formation
Main feature: compromise =⇒ Boltzmann-like collision operatorsMany phenomena taken into account: self-thinking, voting process,contradictory people, leaders, propaganda through media, multipartitedemocracy
Kindymo pluridisciplinary project (PEPS CNRS HuMaIn) withS. Galam, J.-P. Nadal, A. Vignes...
LB, F. Salvarani. Modelling opinion formation by means of kineticequations, in Mathematical modeling of collective behaviour insocio-economic and life sciences, G. Naldi, L. Pareschi, G. Toscanieditors, Springer-Verlag, 2010.
L. Boudin (UPMC & Inria) Mathematical models for opinion dynamics Jussieu, 44 / 44
SociophysicsCompromise vs. self-thinking (with F. Salvarani)ModelExistence result and proofNumerical experiments
Contradictory agents (with A. Mercier, F. Salvarani)Media influence, multipartite system (with R. Monaco, F. Salvarani)Multidimensional modelMathematical resultTwo-party numerical experimentsThree-party numerical experiment
Conclusion and prospects
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