Multi-agent systems modeling opinion formation.

Anna Chiara Lai

Universita degli Studi di Padova

Seminario di Modellistica Differenziale Numerica

March 12, 2013

(joint work with M. Caponigro and B. Piccoli)

Outline

Some opinion formation models

Multi-agent systems

Number of agents goes to infinity:I KTAP approach to socio-economical systems;I A mean field model for consesus problems.

What’s new...

A multi-agent system on the sphereI description of the model and motivations;I preliminary results and numerical simulations.

Outline

Some opinion formation models

Multi-agent systems

Number of agents goes to infinity:I KTAP approach to socio-economical systems;I A mean field model for consesus problems.

What’s new...

A multi-agent system on the sphereI description of the model and motivations;I preliminary results and numerical simulations.

Outline

Some opinion formation models

Multi-agent systems

Number of agents goes to infinity:I KTAP approach to socio-economical systems;I A mean field model for consesus problems.

What’s new...

A multi-agent system on the sphereI description of the model and motivations;I preliminary results and numerical simulations.

Multi-agents opinion formation models

General settings

Network of n agents endowed with an opinion (state of thesystem);

Each agent’s opinion is influenced by other agents (dynamics).

Goal(s)

Describe/predict/control the behaviour of the system;

Investigation of possible emergent behaviours

Discrete-time, state-independent opinion formationmodel

Every agent updates its/his opinion by taking a convex combination ofother agents’ opinion.

Evolution of agent opinions

xi(t+ 1) =n∑j=1

aij(t)xj(t)

where

xi(t) is i-th agent opinion;

the interaction matrix At := (aij(t)) is a stochastic matrix(non-negative elements and At1 = 1).

Network of agents

Interaction matrix At is associated to a directed graph Gt: vertices iand j are connected if aij > 0.

Convergence of opinion dynamics depends on the connectivity of Gt.

Example

Figure: G0, G1 and G2 are not connected, but vertex 1 can reach all othervertices in time [0, 2].

Convergence and consensus

Theorem

Let (At) be a sequence of n× n stochastic matrixes and (Gt) theassiciated sequence of graphs. For every initial condition x(0) ∈ Rn

there exists x∗ ∈ R such that the sequence x(t) solution of

xi(t+ 1) =n∑j=1

aij(t)xj(t)

converges exponentially to 1x∗ if

a) All non-zero elements (aij(t)) are uniformly bounded from below;

b) There exists a sequence of contiguous (time) intervals of boundedlength over each of which at least a vertex is connected to all others.

Weaking connectivity: convergence without consensus

Theorem

If for every t (Gt) is symmetric and aii(t) > 0 and if

a) all non-zero elements (aij(t)) are uniformly bounded from below;

the convergence to a limit and consensus is reached in every connectedcomponent.

Continuous-time opinion, state-independent dynamics

xi(t) =

m∑j=1

aij(t)(xj(t)− xi(t))

or in compact formx(t) = L(t)x(t)

where L(t) = (lij(t)) has zero rows sums and nonnegative off-diagonalelements. In particular

lij =

{aij if i 6= j

−∑n

j=1 aij otherwise

Remark. Consensus is reached when L satisfies connectivity conditionssimilar to the discrete-time case.Open problem. Establish sufficient conditions for convergence withoutconsensus.

State-dependent interactions: Krause model

Bounded confidence

Every agent averages its opinion with close agents and ignores faropinions.

Discrete-time

xi(t+ 1) =

∑j:|xi(t)−xj(t)|<ε xj(t)

|{j : |xi(t)− xj(t)| < εi}|(1)

Continuous-time

xi(t) =∑

j:|xi(t)−xj(t)|<ε

xj(t)− xi(t) (2)

Matrix formulation: x(t) = A(x(t))x(t).

Example

Some properties in continuous-time case 1/2

Weighted system:

xi(t) =∑

j:|xi(t)−xj(t)|<ε

wj(xj(t)− xi(t)) (3)

Convergence. Let x∗i = limt→∞ xi(t). If x evolves according to (4) theselimits are well defined and for every i, j

x∗i = x∗j |x∗i − x∗j | > ε.

1Blondel, V. D., Hendrickx, J. M., Tsitsiklis, J. N. “Continuous-time average-preserving

opinion dynamics with opinion-dependent communications”. SIAM Journal on Control andOptimization, 48(8), 5214-5240. (2010).

Some properties in continuous-time case 2/2

Weighted system:

xi(t) =∑

j:|xi(t)−xj(t)|<ε

wj(xj(t)− xi(t)) (4)

Clusters. The set of agents tending to a common limit x∗A is calledcluster. The weight WA of a cluster A is the sum of the weight of itsagents.Stability. The system is stable w.r.t. small perturbations (i.e. an extraagent with arbitrary small weight is introduced) if and only if all theclusters are far enough, in particular for every couple of clusters A andB

1 +min{WA,WB}max{WA,WB}

.

1Blondel, V. D., Hendrickx, J. M., Tsitsiklis, J. N. “Continuous-time average-preserving

opinion dynamics with opinion-dependent communications”. SIAM Journal on Control andOptimization, 48(8), 5214-5240. (2010).

Quick overview on state-dependent systems

Dx(t) = A(x(t))x(t)

(D discrete or continuous derivative)

long time behaviour is in general hard/impossible to study;

Krause model is characterized by clustering phenomena;

number of clusters, inter-cluster distance, dependence on initialdata are still active research domains.

Kinetic model of population dynamics

n interacting populations, with state xi(t) ∈ [0, 1];probability density fi(t, x) so that

Pi(t, x ∈ [x1, x2]) =

∫ x2

x1

fi(t, x)dx

encounter rate νi,j(x, y)switch density ψi,j(z, y;x)

∂fi∂t

(t, u) =

n∑j=1

∫ 1

0

∫ 1

0

νij(y, z)ψij(y, z;x)fi(t, y)fj(t, z)dydz (inflow)

− fi(t, x)

n∑j=1

∫ 1

0

νij(x, y)fj(t, y)dy (outflow)

1L. Arlotti, N. Bellomo, “Solutions of a new class of nonlinear kineticmodels of population dynamics”, Appl. Math. Lett., 9 (2), 65-70, 1996

Theoretical results1 and applications2

extension to triple encounters;

existence and uniqueness of solution under standard assumptions;

existence and preliminary study of equilibria.

applications to complex socio-economical systems (welfare politics,democratization of a dictatorship, competition for a secession)

1L. Arlotti, N. Bellomo, “Solutions of a new class of nonlinear kineticmodels of population dynamics”, Appl. Math. Lett., 9 (2), 65-70, 1996

2Marsan, G. A., Bellomo, N., Egidi, M. (2008). Towards a mathematicaltheory of complex socio-economical systems by functional subsystemsrepresentation. Kinetic and Related Models, 1, 249-278.

Kinetic model of opinion formation

x′ = x− γP (|x|)(x− x∗) + νD(|x|)

x′∗ = x∗ − γP (|x∗|)(x∗ − x) + ν∗D(|x∗|)

x and x∗ opinions

compromise terms: γ > 0 compromise propensity; P (|x|) localrelevance for compromise (extremal values penalized);

diffusion terms: ν, ν∗ random variables (zero mean, σ2 variance,values in B ⊂ R); D(|x|) local relevance for diffusion (extremalvalues penalized);

∂f

∂t=

∫B2

∫I

(′β

1

Jf(′x)f(′x∗)− βf(x)f(x∗)

)dx∗dνdν∗

1G. Toscani, “Kinetic models of opinion formation”, Commun. Math.Sci. 4 (3) (2006) 481-496

A consensus problem based on mean field games

Large population stocastic consesus problem

Stochastic dynamics

dzi(t) = ui(t)dt+ σdwi(t)

Long Run Average cost function

J(N)i = lim sup

T→∞

1

T

∫ T

0

zi − 1

N − 1

N∑k 6=i

zk

2

+ ru2i

dt

1Nourian, M., Caines, P. E., Malhame, R. P., Huang, M. “A solution tothe consensus problem via stochastic mean field control”. In 2nd IFACNecSys Workshop, Annecy, France (pp. 323-328). 2010

A consensus problem based on mean field games

Mean field equation system

ds

dt=

1√rs+ z∗

dzαdt

= − 1√rzα −

1

rs α ∈ A

z∗(t) =

∫Azα(t)dF (α)

where

z∗(t) = limN→∞

1

N

N∑i=1

Ezi a.s. dF

zα = Ezα

1Nourian, M., Caines, P. E., Malhame, R. P., Huang, M. “A solution to the consensus

problem via stochastic mean field control”. In 2nd IFAC NecSys Workshop, Annecy, France(pp. 323-328). 2010

A consensus problem based on mean field games

Unique, explicit solution, in particular s(·) and z∗(·) are constant;

the associated control law yields a mean consensus in thestochastic game;

extension to discounted cost functions and to cooperative systems.

every agent is required to have a-priori knowledge of the initialstate of the system.

1Nourian, M., Caines, P. E., Malhame, R. P., Huang, M. “A solution tothe consensus problem via stochastic mean field control”. In 2nd IFACNecSys Workshop, Annecy, France (pp. 323-328). 2010

A consensus problem based on mean field games

Unique, explicit solution, in particular s(·) and z∗(·) are constant;

the associated control law yields a mean consensus in thestochastic game;

extension to discounted cost functions and to cooperative systems.

every agent is required to have a-priori knowledge of the initialstate of the system.

1Nourian, M., Caines, P. E., Malhame, R. P., Huang, M. “A solution tothe consensus problem via stochastic mean field control”. In 2nd IFACNecSys Workshop, Annecy, France (pp. 323-328). 2010

An opinion formation model on Sd

The model

Model type: fixed topology -continuous-time;

Interactions: both attractive andrepulsive;

Agent space: Sd.

The system

xi =∑i 6=j

aij(xj − 〈xi, xj〉xi)

Remark. Classical system xi =∑

i 6=j aij(xj − xi) with aij > 0.

Equilibria

consensus (aka rendez-vous): xi = xj for every i, j;

antipodal: xi = ±xj for every i, j and xi = −xj for some i, j;

poligonal.

In general xi = 0 ifxi = ciαi (5)

for some ci ∈ R, where αi =∑n

j=1 aijxj is the total influence on xi.

A stability result.Fix an equilibrium x∗ = (x∗i ) and for every i consider the system

xi =

n∑i=1

aij(x∗j − 〈x∗j , xi〉xi).

If ci > 0 then the equilibrium is stable;

if ci < 0 then the equilibrium is unstable.

Asymptotic behaviour

Theorem

If the interaction matrix is symmetric then the system converges.

Hint of proof: The integral of the kinetic energy of the system can bewritten explicitely and it is a bounded function.

Figure: Kinetic energy of a system of 150 agents with symmetric adjacencymatrix

Energy decay in sign-symmetric matrices

Figure: 10 agents, sign-symmetric matrix

Energy decay in general matrices

Figure: 10 agents, random matrix

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