Mathematical models for opinion dynamics · 2014. 1. 24. · Mathematical models for opinion dynamics Laurent Boudin Lab. Jacques-Louis Lions, Univ. Pierre et Marie Curie & Reo, Inria

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

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


Sociophysics

Outline

1 Sociophysics






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.


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.


Compromise vs. self-thinking

Outline

1 Sociophysics






Compromise vs. self-thinking Model

Outline

1 Sociophysics







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)



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.



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).



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.



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 .



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.


Compromise vs. self-thinking Existence result and proof

Outline

1 Sociophysics







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.



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.



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.



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.



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∞ .


Compromise vs. self-thinking Numerical experiments

Outline

1 Sociophysics







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.)



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%.



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.



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.


Contradictory agents

Outline

1 Sociophysics







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)



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∗



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



Disjoint-supported initial data (2)

I+(f + g) I+(f )


Media influence, multipartite system

Outline

1 Sociophysics






Media influence, multipartite system Multidimensional model

Outline

1 Sociophysics







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)



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



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


Media influence, multipartite system Mathematical result

Outline

1 Sociophysics






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.


Media influence, multipartite system Two-party numerical experiments

Outline

1 Sociophysics







Influence of one unique media 1/2

Media opinion: (0.9, 0.9), strength: α1 = 0.1β

Situation 1 – Concentration effect.



Influence of one unique media 2/2

Media opinion: (0.9, 0.9), strength: α1 = 0.1β

Situation 2 – Threshold effect.



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


Media influence, multipartite system Three-party numerical experiment

Outline

1 Sociophysics






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


Conclusion and prospects

Outline

1 Sociophysics








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.


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


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Mathematical models for opinion dynamics · 2014. 1. 24. · Mathematical models for opinion dynamics Laurent Boudin Lab. Jacques-Louis Lions, Univ. Pierre et Marie Curie & Reo, Inria