Kinetic models of opinion formationmate.unipv.it/toscani/Talks/Part5.pdf · possibilities (among...

OutlinesSociophysical modelling

The quasi-invariant opinion limitConclusions

Kinetic modelsof opinion formation

Giuseppe Toscani

Department of Mathematics

University of Pavia, Italy

Porto Ercole, June 8-10 2008

Summer School METHODS AND MODELS OF KINETIC THEORY

Giuseppe Toscani Kinetic models of opinion



Outline

1 Sociophysical modellingIntroductionModeling interactionsKinetic equations

2 The quasi-invariant opinion limitTowards a simpler modelFokker-Planck equations of opinion formation

Steady states

3 Conclusions




IntroductionModeling interactionsKinetic equations

Collective phenomena

Microscopic models of both social and political phenomenadescribing collective behaviors and self–organization in a societyrecently analyzed by several authors[Galam S., Gefen Y., Shapir Y. (1982), Ochrombel R. (2001),Stauffer D. (2002), Stauffer D., de Oliveira P.M.C. (2002),Sznajd-Weron K., Sznajd J. (2000)].Books by [Liggett T.M. Stochastic interacting systems 1999,Weidlich W. Sociodynamics 2000].

The collective behaviors of a society composed by a sufficiently largenumber of individuals (agents) described using the laws of statisticalmechanics as it happens in a physical system composed of manyinteracting particles.

The details of the social interactions between agents thencharacterize the emerging statistical phenomena.





Opinion formation

Modeling of opinion formation attracted the interest of aincreasing number of researchers[Deffuant G., Amblard F., Weisbuch G., Faure T. (2002),Liggett T.M. Stochastic interacting systems 1999,Ochrombel R. (2001), Sznajd-Weron K., Sznajd J. (2000)]

The starting point of a large part of these models, however, isrepresented by a cellular automata, where the lattice pointsare the agents.

Any of the agents of a community is initially associated with arandom distribution of numbers, one of which is the opinion.

Society is modelled as a graph, where each agent interactswith his neighborhoods in iterative way.





Compromise process

Other attempts have been successfully applied[Ben-Naim E. (2005), Slanina F. Lavicka H. (2003)].Describe formation of opinion by means of mean fields modelequations.In general partial differential equations of diffusive type, treatedanalytically to give explicit steady states.[Ben-Naim E. (2005)] focused on two aspects of opinionformation, which could be responsible of the formation ofcoherent structures.The compromise process, in which pairs of agents reach a faircompromise after exchanging opinions[Ben-Naim E., Krapivski P.L., Redner S. (2003),Deffuant G., Amblard F., Weisbuch G., Faure T. (2002),Fortunato S. (2004), Hegselman R., Krause U. (2002),Weisbuch G., Deffuant G., Amblard F., Nadal J.P. (2002)].





Diffusion process

The diffusion process, which allows individual agents tochange their opinions in a random diffusive fashion.

The compromise process has is basis on the human tendencyto settle conflicts.

Diffusion accounts for the possibility that people may changeopinion through a global access to information.

Aspect gaining in importance due to the emerging of newpossibilities (among them electronic mail and web navigation[Rash W. Politics on the nets: wiring the political process 1997]).

We consider here a class of kinetic models of opinionformation, bases on two-body interactions involving bothcompromise and diffusion properties in exchanges betweenindividuals.





Equilibria

The kinetic model gives in a suitable asymptotic limit(hereafter called quasi-invariant opinion limit) a partialdifferential equation of Fokker-Planck type for the distributionof opinion among individuals.

Similar diffusion equations in [Slanina F. Lavicka H. (2003)]as the mean field limit of the Ochrombel simplification of theSznajd model [Sznajd-Weron K., Sznajd J. (2000)].

The equilibrium state of the Fokker-Planck equation can becomputed explicitly and reveals formation of picks incorrespondence to the points where diffusion is missing.

The mathematical methods close to kinetic theory of granulargases, where the limit procedure is known as quasi-elasticasymptotics[McNamara S., Young W.R. (1993), Toscani G. (2000)].





Related results

Similar analysis on a kinetic model of a simple marketeconomy with a constant growth mechanism[Cordier S., Pareschi L., Toscani G. (2005),Slanina F. (2004)].

Formation of steady states with Pareto tails[Pareto V. Cours d’Economie Politique 1897].

Mean field approximation leads to the same Fokker-Planckequation [J.P.Bouchaud, M.Mezard (2000),Malcai O., Biham O., Solomon S., Richmond P. (2002),Solomon S. (1998)], showing consistency between kinetic andstochastic approaches.





Trades I

The goal of the forthcoming kinetic model of opinion formation,is to describe the evolution of the distribution of opinions in asociety by means of microscopic interactions among agents orindividuals which exchange information.

We associate opinion with a variable w which varies continuouslyfrom −1 to 1, where −1 and 1 denote the two (extreme) oppositeopinions. We moreover assume that interactions do not destroy thebounds, which corresponds to impose that the extreme opinions cannot be crossed.

Denote by (w ,w∗), with w ,w∗ ∈ I the pair of opinions of twoindividuals before the interaction and (w ′,w ′

∗) their opinionsafter exchanging information between them and with theexterior.





Exchange of opinion

Let I = [−1,+1] denote the interval of possible opinions. Wedescribe the binary interaction by the rules

w ′ = w − γP(|w |)(w − w∗) + ηD(|w |),

w ′∗ = w∗ − γP(|w∗|)(w∗ − w) + η∗D(|w∗|).

Opinions can not cross boundaries. The interaction takes placeonly if both w ′,w ′

∗ ∈ I.γ ∈ (0, 1/2) is a given constant. η and η∗ random variables withthe same distribution with variance σ2 and zero mean, takingvalues on a set B ⊆ IR.The constant γ and the variance σ2 measure respectively thecompromise propensity and the modification of opinion due todiffusion.The functions P(·) and D(·) describe the local relevance of thecompromise and diffusion for a given opinion.





Details of binary exchange

The first part is related to the compromise propensity of theagents

The second contains the diffusion effects of external events.

The pre-interaction opinion w increases (getting closer to w∗)when w∗ > w and decreases in the opposite situation.

The presence of both the functions P(·) and D(·) is linked tothe hypothesis that the availability to the change of opinion islinked to the opinion itself, and decreases as soon as one getcloser to extremal opinions.

Extremal opinions are more difficult to change.

Assume P(|w |) and D(|w |) non increasing with respect to|w |, and in addition 0 ≤ P(|w |) ≤ 1, 0 ≤ D(|w |) ≤ 1.





Particular cases

In absence of the diffusion contribution (η, η∗ ≡ 0),

w ′ + w ′∗ = w + w∗ + γ(w − w∗) (P(|w |)− P(|w∗|))

w ′ − w ′∗ = (1− 2γ(P(|w |) + P(|w∗|))) (w − w∗).

If P(·) is not constant, P = 1, the total momentum is notconserved and it can increase or decrease depending on theopinions before the interaction.If P(·) is assumed constant, the interaction correspond to agranular gas like interaction (or to a traffic flow model[Klar A., Wegener R. (1996)])In a single interaction, the compromise propensity implies thatthe difference of opinion is diminishing, with|w ′−w ′

∗| = (1− 2γ)|w −w∗|. Thus all agents will end up in thesociety with exactly the same opinion.





Particular cases II

In this elementary case a constant part of the relative opinion isrestituted after the interaction.In the other cases

|w ′ − w ′∗| = (1− 2γ(P(|w |) + P(|w∗|))) |w − w∗|.

This implies

0 ≤ ε(w ,w∗) = 1− 2γ(P(|w |) + P(|w∗|)) ≤ 1.

The general case corresponds to a granular gas interaction witha variable coefficient of restitution [Toscani G. (2000)].In absence of diffusion, the lateral bounds are not violated

w ′ = (1− γP(|w |))w + γP(|w |)w∗





The Boltzmann equation

Let f (w , t) denote the distribution of opinion w ∈ I at timet ≥ 0.

Standard methods of kinetic theory[Cercignani et al.The mathematical theory of dilute gases 1994]allow to describe the time evolution of f as a balance betweenbilinear gain and loss of opinion terms

∂f

∂t=

∫B2

∫I

(′β

1

Jf (′w)f (′w∗)− βf (w)f (w∗)

)dw∗ dη dη∗

(′w ,′ w∗) are the pre-interaction opinions that generate thecouple (w ,w∗) of opinions after the interaction.

The kernels ′β and β are related to the details of the binaryinteraction.





The Boltzmann equation II

The transition rate is taken of the form

β(w ,w∗)→(w ′,w ′∗) = Θ(η)Θ(η∗)χ(|w ′| ≤ 1)χ(|w ′

∗| ≤ 1),

χ(A) is the indicator function of the set A, and Θ(·) is asymmetric probability density with zero mean and variance σ2.

The rate function β(w ,w∗)→(w ′,w ′∗) characterizes the effects of

external events on opinion through the distribution of therandom variables Θ and Θ∗

The support B of the symmetric random variable is a subsetof I, to prevent diffusion to generate a complete change ofopinion.





Simplifications

The main problem in opinion dynamics relies in looking for theformation of stationary profiles for the opinion.

In the kinetic picture this corresponds to look for the largetime behavior of the density of opinion f (w , t).

Extremely difficult problem to describe in details thelarge-time behavior of the solution for a general kernel.

Restrict the analysis to the cases in which the kernel β doesnot depend on the opinion variables (Maxwellian case).

Any choice of D(|w |) and B which preserve the lateral boundsof extreme opinions allows to study in details the dynamics ofthe model with a significant simplification.





Simplifications II

The Maxwellian assumption implies

β(w ,w∗)→(w ′,w ′∗) = β(η, η∗) = Θ(η)Θ(η∗).

In this case the weak form reads

d

dt

∫I

φ(w)f (w , t) dw =

⟨∫I2

f (w)f (w∗)(φ(w ′)− φ(w))dw∗dw

⟩Conservation of the total opinion is obtained for φ(w) = 1,which represents in general the only conservation propertysatisfied by the system.

The choice φ(w) = w is of particular interest since it givesthe time evolution of the average opinion.





Evolution of moments

We have

d

dt

∫I

wf (w , t) dw =

⟨∫I2

f (w)f (w∗)γ(P(|w |)w∗ − P(|w |)w)dw∗dw

⟩

+

⟨∫I2

f (w)f (w∗)ηD(|w |)dw∗dw

⟩In case P(|w |) = 1, the first contribution disappears. Bysymmetry ⟨∫

I2

f (w)f (w∗)γ(w∗ − w)dw∗dw

⟩= 0.





Evolution of average opinion

Since 〈η〉 = 0 implies

d

dt

∫I

wf (w , t) dw =

= 〈η〉∫I2

χ(|w ′| ≤ 1)χ(|w ′∗| ≤ 1)D(|w |)f (w)f (w∗)dw∗dw = 0.

P constant implies conservation of the average opinion.In general

d

dt

∫I

wf (w , t) dw

= γ

∫I

P(|w |)f (w) dw

∫I

wf (w) dw − γ

∫I

wP(|w |)f (w) dw .

The evolution of the average opinion is not closed.





Evolution of other moments

Fix now φ(w) = w2.

d

dt

∫I

w2f (w , t) dw

= γ2

∫I2

P(|w |)2(w − w∗)2f (w)f (w∗) dwdw∗

−2γ

∫I2

P(|w |)w(w − w∗)f (w)f (w∗) dwdw∗ + σ2

∫I

D(|w |)2f (w)dw .

Choosing P(|w |) = 1, m the constant value of the averageopinion

d

dt

∫I

w2f (w , t) dw

= −2γ(1− γ)

[∫I

w2f (w) dw −m2

]+ σ2

∫I

D(|w |)2f (w)dw .




Towards a simpler modelFokker-Planck equations of opinion formation

Steady states

Towards a simpler model

Previous analysis shows that in general it is quite difficultboth to study in details the evolution of the opinion density,and to describe its asymptotic behavior.

For a general kernel the mean opinion is varying in time.

Asymptotics of the equation result in simplified models(generally of Fokker-Planck type)

Particularly relevant in case of a good approximation of thestationary profiles of the kinetic equation.

Physical basis to these asymptotics comes from theinteraction rule.

Assume P(|w |) = 1, so that conservation both of mass andmomentum holds.





Steady states

Simpler models II

The interaction rule has following properties

〈w ′ + w ′∗〉 = w + w∗, 〈w ′ − w ′

∗〉 = (1− 2γ)(w − w∗).

The first equality is the property of mean conservation of opinion.The second refers to the compromise propensity, which plays infavor of the decrease (in mean) of the distance of opinions afterthe interaction.Same property in a collision between molecules in a granular gas.There e = 2γ is called coefficient of restitution[Toscani G. (2000)].Suppose most of the interactions produce a very small exchangeof opinion (γ → 0), while mean properties remain true at amacroscopic level.





Steady states

Simpler models II

While γ → 0

⟨∫I2

(w ′ + w ′∗)f (w)f (w∗)dwdw∗

⟩= 2

∫I

wf (w)dw = 2m(t)

remains constant,and

1

2

⟨∫I2

(w ′ − w ′∗)

2f (w)f (w∗)dwdw∗

⟩=

∫I

w2f (w) dw −m20 = Cf (t)

varies with time.Then m(t) = m0 independently of the value of γ. Moreover

dCf (t)

dt= −2γ(1− γ)Cf (t) + σ2

∫I

D(|w |)2f (w)dw .





Steady states

Scaling

Set

τ = γt, g(w , τ) = f (w , t).

Then f0(w) = g0(w), and

dCg (τ)

dτ= −2 (1− γ) Cg (τ) +

σ2

γ

∫I

D(|w |)2f (w)dw .

Let γ → 0 and σ → 0 but σ2/γ = λ,

dCg (τ)

dτ= −2Cg (τ) + λ

∫I

D(|w |)2f (w)dw .

t = τ/γ, and γ → 0 describes the large-time behavior of f (v , t).Since f (w , t) = g(w , τ) the large-time behavior of f (w , t) isclose to the large-time behavior of g(w , τ).





Steady states

Mathematical assumptions

Let

Mp(A) =

{Θ ∈M0 :

∫A|w |pdΘ(w) < +∞, p ≥ 0

}the space of all Borel probability measures of finite momentumof order p, equipped with the topology of the weak convergenceof the measures.Let Fs(I), be the class of all real functions h on I such thath(±1) = h′(±1) = 0, and h(m)(v) is Holder continuous of orderδ,

‖h(m)‖δ = supv 6=w

|h(m)(v)− h(m)(w)||v − w |δ

< ∞,

the integer m and the number 0 < δ ≤ 1 are such thatm + δ = s, and h(m) denotes the m-th derivative of h.





Steady states

Scaling II

The scaled density g(v , τ) = f (v , t) satisfies the equation

d

dτ

∫I

g(w)φ(w) dw =1

γ

⟨∫I2

g(w)g(w∗)(φ(w ′)− φ(w))dw∗dw

⟩.

Set φ ∈ F2+δ(I).By the interaction rule

w ′ − w = γ(w∗ − w) + ηD(|w |) � 1.

Use a second order Taylor expansion of φ around w

φ(w ′)− φ(w) = (γ(w∗ − w) + ηD(|w |))φ′(w)

+1

2(γ(w∗ − w) + ηD(|w |))2 φ′′(w).





Steady states

Scaling II

Inserting this expansion in the collision operator

d

dτ

∫I

g(w)φ(w) dw =

⟨1

γ

∫I2

[(γ(w∗ − w) + ηD(|w |))φ′(w)

+1

2(γ(w∗ − w) + ηD(|w |))2φ′′(w)]g(w)g(w∗)dw∗ dw

⟩+ R(γ, σ)

R is the remainder

R(γ, σ) =1

2γ

⟨∫I2

(γ(w∗ − w) + ηD(|w |))2·

·(φ′′(w)− φ′′(w)

)g(w)g(w∗) dw∗ dw

⟩.





Steady states

Scaling III

One shows that R(γ, σ) converges to zero as as both γ and σconverge to zero, in such a way that σ2 = λγ.Within the same scaling

limγ→0

1

γ

⟨∫I2

[(γ(w∗ − w) + ηD(|w |))φ′(w) +

+1


⟩=

∫I

[(m − w)φ′(w) +

λ

2D(|w |)2φ′′(w)

]g(w)dw

Since φ ∈ Fs(I), weak form of the Fokker-Planck equation

∂g

∂τ=

λ

2

∂2

∂w2

(D(|w |)2g

)+

∂

∂w((w −m)g).





Steady states

Main result

We proved

Theorem

Let the probability density f0 ∈M0(I), and let the symmetricrandom variable Y which characterizes the kernel have a density inM2+α, with α > δ. Then, as γ → 0, σ → 0 in such a way thatσ2 = λγ the weak solution to the Boltzmann equation for thescaled density gγ(v , τ) = f (v , t), with τ = γt converges, up toextraction of a subsequence, to a probability density g(w , τ). Thisdensity is a weak solution of the Fokker-Planck equation, and it issuch that the average opinion is conserved.





Steady states

Other Fokker-Planck models

Previous Theorem can be extended to interaction rules with ageneral function P(|w |).The main difference is the evaluation of the first order term∫

I2

P(|w |)(w∗ − w)φ′(w)g(w)g(w∗)dw∗ dw

Let m(τ) be the value of the average opinion at time τ ≥ 0

m(τ) =

∫I

wg(w , τ)dw .

The scaling t → τ implies

dm(τ)

dτ= m(τ)

∫I

P(|w |)g(w , τ) dw −∫I

wP(|w |)g(w , τ) dw .





Steady states

Other Fokker-Planck models II

As γ → 0 we obtain that g(w , τ) satisfies the Fokker-Planckequation

∂g

∂τ=

λ

2

∂2

∂w2

(D(|w |)2g

)+

∂

∂w(P(|w |)(w −m(t))g).

The balance γ → 0 and σ → 0 in such a way that σ2/γ = λ,allows to recover in the limit the contributions due both tocompromise propensity and the diffusion.

Other limits can be considered, which are diffusion dominated(σ2/γ = ∞) or compromise dominated(σ2/γ = 0).

However, the formation of an asymptotic profile for theopinion is linked to the first balance [Ben-Naim E. (2005)].





Steady states

Other models

Pure diffusion and drift equations connected to this matter havebeen recently introduced in [Slanina F. Lavicka H. (2003)].These equations, in our picture, refer to diffusion dominated(σ2/γ = ∞) or compromise dominated (σ2/γ = 0) limits.The diffusion dominated limit takes into account only thesecond-order term into the Taylor expansion.Suppose

σ2

γα→ λ , α < 1.

Then

τ = γαt, g(w , τ) = f (w , t)





Steady states

Other models

We obtain

d

dτ

∫I

g(w)φ(w) dw =

⟨1

γα

∫I2

[(γ(w∗ − w) + ηD(|w |))φ′(w)

+1


⟩+ R(γ, σ)

Since α < 1, the first order term in the Taylor expansionvanishes in the limit, and g satisfies the diffusion equation

∂g

∂τ=

λ

2

∂2

∂w2

(D(|w |)2g

).

Set D(|w |) =√

1− w2, λ = 2. Then

∂g

∂τ=

∂2

∂w2

[(1− w2)g

].





Steady states

Other models II

The compromise dominated (σ2/γ = 0) limit corresponds to thescaling

σ2

γα→ λ , α > 1.

In this case, we scale as

τ = γt, g(w , τ) = f (w , t).

The diffusion part disappears in the limit and we obtain the puredrift equation

∂g

∂τ=

∂

∂w(P(|w |)(w −m(t))g).





Steady states

Other models III

The choice P(|w |) = 1− w2 has been considered in[Slanina F. Lavicka H. (2003)].In this case [Aletti G., Naldi G., Toscani G. (2007)]

∂g

∂τ=

∂

∂w

((1− w2)(w −m(t))g

).

m(τ) satisfies

dm(τ)

dτ= −m(τ)

∫I

w2g(w , τ) dw +

∫I

w3g(w , τ) dw .

Related equations in [Slanina F. Lavicka H. (2003),Sznajd-Weron K., Sznajd J. (2000)] in case of two opinions

∂g

∂τ= ± ∂

∂w

((1− w2)wg

).





Steady states

Steady states

Explicit steady states if P(|w |) = 1, which implies conservation ofthe average opinionSet D(|w |) = 1− w2. The steady state distribution of opinion is asolution to

λ

2

∂

∂w

((1− w2)2g

)+ (w −m)g = 0

We obtain

g∞(w) = cm,λ (1− w)−2+m/(2λ) (1 + w)−2−m/(2λ) exp

{− 1−mw

λ(1− w2)

}.

The constant cm,λ is such that the mass of g∞ is equal to one.Note that g∞(±1) = 0. The solution is regular, but not symmetricunless m = 0. Two picks (on the right and on the left of zero).





Steady states

Steady states II

Similar result expected from the choice D(|w |) = 1− |w |.The steady state distribution of opinion is a solution to

λ

2

∂

∂w

((1− |w |)2g

)+ (w −m)g = 0.

We obtain

g∞(w) = cm,λ (1− |w |)−2−2/λ exp

{−1−mw/|w |

2λ(1− |w |)

}.

Low regularity of D(|w |) reflects on the steady solution, whichhas a jump in w = 0.As in the first case, the presence of the exponential assures thatg∞(±1) = 0.





Steady states

Steady states III

Last, consider D(|w |) =√

1− w2.The steady state distribution of opinion is a solution to

λ

2

∂

∂w

((1− w2)g

)+ (w −m)g = 0.

We obtain

g∞(w) = cm,λ

(1

1 + w

)1−(1+m)/λ (1

1− w

)1−(1−m)/λ

.

Since −1 < m < 1, g∞ is integrable on I.Differently from the previous cases, however g∞(w) tends toinfinity as w → ±1, and it has no peaks inside the interval I.Extreme opinions win!




Conclusions

Various kinetic models of opinion formation, based on binaryinteractions involving both compromise and diffusion propertiesin exchanges between individuals have been introduced.A suitable scaling of compromise and diffusion allows to deriveFokker-Planck equations for which it is easy in many cases torecover explicitly the stationary distribution of opinion.Among these Fokker-Planck equations, one is emerging andtakes the role of the analogous one obtained in[J.P.Bouchaud, M.Mezard (2000),Cordier S., Pareschi L., Toscani G. (2005)] for the evolution ofwealth.The main feature of this equation is that moments can beevaluated in closed form, and possible models of hydrodynamicscan eventually follow as in classical kinetic theory of rarefiedgases.


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