Multi-agent Cooperative Dynamical Systems: Theory and ... · Multi-agent Cooperative Dynamical...

Multi-agent Cooperative Dynamical Systems:Theory and Numerical Simulations

C. Canuto, F. Fagnani and P. TilliDipartimento di Matematica - Politecnico di Torino

IMA Workshop on Novel Discretization TechniquesMinneapolis, MN, November 1st 2010

Claudio Canuto () Consensus models and their simulation 1 / 39

Outline

1 Cordinated control problems

2 The continuous-in-space models

3 Theoretical results

4 A numerical algorithm

5 Numerical simulations

6 Perspectives


Introduction

Problems of coordinated control deal with the behaviour of networks ofindividuals (’the agents’), who can exchange partial information among them, andwho aim at reaching or preserving a particular configuration.

Examples of such situations are:

Rendez-vous

Formation

Flocking

These problems are also referred to as reaching a consensus, or emerging of acommon belief, without a central leadership.

The mathematical modeling of such situations is a challenging task, with a widerange of applications.


Introduction



Rendez-vous

Formation

Flocking




Introduction



Rendez-vous

Formation

Flocking




Introduction



Rendez-vous

Formation

Flocking




General setting

Assume we are given a set A of agents. The set A has large cardinality N .

The position of agent j ∈ A at time t will be denoted by xj(t); it belongs tosome Euclidean space Rq.In a different context, one can think of xj(t) as the opinion of the member j of acommunity A.

We will assume that all positions undergo the same dynamics governed by a linearinput/output law which can be in either discrete or continuous time.

Note the Lagrangean point of view assumed in such a description.


General setting






General setting






Discrete-in-time models

The evolution of all agents takes place over the lattice 0, τ, 2τ, . . . where τ > 0 isa fixed time step.For the sake of notational simplicity, we assume τ = 1, so that each xj(t) dependsupon t ∈ N.

We assume a first-order evolution law of every agent:

xj(t+ 1) = xj(t) + uj(t) .

The vector uj(t) ∈ Rq plays the role of a control input function that each agentcan autonomously choose on the basis of the information available at time t.

In general, it will be feedback control, i.e., a function of the position xj(t) aswell of the information transmitted by its neighbors.

The analogous continuous-in-time models take the form

xj(t) = uj(t) , t ∈ R+ ,

indicating that the control variable is just the velocity of the agent.





xj(t+ 1) = xj(t) + uj(t) .




xj(t) = uj(t) , t ∈ R+ ,






xj(t+ 1) = xj(t) + uj(t) .




xj(t) = uj(t) , t ∈ R+ ,



Consensus models

The dynamics of a consensus model is given by a law of the form

xj(t+ 1) = xj(t) +∑k∈V

Pjk (xk(t)− xj(t)) , j ∈ A ,

where P is a sub-stochastic matrix, i.e., it satisfies

Pjk ≥ 0 ,∑k

Pjk ≤ 1 .

The matrix P may have either deterministic or stochastic entries, and may or maynot depend on time.In many situations of interest, Pjk = Pjk(t) depends on time via the positionsxj(t) and xk(t).


Consensus models

The dynamics of a consensus model is given by a law of the form

xj(t+ 1) = xj(t) +∑k∈V

Pjk (xk(t)− xj(t)) , j ∈ A ,

where P is a sub-stochastic matrix, i.e., it satisfies

Pjk ≥ 0 ,∑k

Pjk ≤ 1 .

The matrix P may have either deterministic or stochastic entries, and may or maynot depend on time.In many situations of interest, Pjk = Pjk(t) depends on time via the positionsxj(t) and xk(t).


Communication Netwoks

The matrix P induces a communication network topology, i.e., at every timeinstant t it defines a communication graph Gt = (A, Et), where Et is the family ofpairs of vertices in A defined by

(j, k) ∈ Et iff Pjk 6= 0 .

Given j ∈ A we consider the neighborhood of j at time t defined as

At(j) = {k ∈ V | (j, k) ∈ Et} ,

whose cardinality will be denoted by Nj(t).


Communication Netwoks

The matrix P induces a communication network topology, i.e., at every timeinstant t it defines a communication graph Gt = (A, Et), where Et is the family ofpairs of vertices in A defined by

(j, k) ∈ Et iff Pjk 6= 0 .

Given j ∈ A we consider the neighborhood of j at time t defined as

At(j) = {k ∈ V | (j, k) ∈ Et} ,

whose cardinality will be denoted by Nj(t).


The Rendez-vous problem

We say that the feedback control law defined by the matrix P satisfies therendez-vous problem if there exists x ∈ Rq such that for every agent j ∈ A andevery initial condition xj(0), one has

limt→+∞

xj(t) = x .

Moreover, we say that the control law satisfies the barycentral rendez-vousproblem if x = N−1

∑j xj(0).

A vaste literature on rendez-vous and similar problems is available, e.g.,

Krause (2000), Lorenz (2005), Blondel, Hendricks and Tsitsiklis (2007), ...

with different points of view and contributions from mathematicians, statisticians,physicists, engineers, social scientists, ...


The Rendez-vous problem

We say that the feedback control law defined by the matrix P satisfies therendez-vous problem if there exists x ∈ Rq such that for every agent j ∈ A andevery initial condition xj(0), one has

limt→+∞

xj(t) = x .

Moreover, we say that the control law satisfies the barycentral rendez-vousproblem if x = N−1

∑j xj(0).

A vaste literature on rendez-vous and similar problems is available, e.g.,

Krause (2000), Lorenz (2005), Blondel, Hendricks and Tsitsiklis (2007), ...

with different points of view and contributions from mathematicians, statisticians,physicists, engineers, social scientists, ...


Krause’s model - I

A limitation in the communication length among agents is modeled by Krauseas follows.Let R > 0 represent the maximum communication length. Define thecommunication graph Gt = (A, Et) by the rule

(j, k) ∈ Et iff |xk(t)− xj(t)| ≤ R .

The corresponding dynamics is defined as

xj(t+ 1) =1

Nj(t)

∑k∈Aj(t)

xk(t) .

This means that at the new time step each agent places itself in the barycenter ofthe position of all agents which it sees in a neighborhood of radius R around it.


Krause’s model - I

A limitation in the communication length among agents is modeled by Krauseas follows.Let R > 0 represent the maximum communication length. Define thecommunication graph Gt = (A, Et) by the rule

(j, k) ∈ Et iff |xk(t)− xj(t)| ≤ R .

The corresponding dynamics is defined as

xj(t+ 1) =1

Nj(t)

∑k∈Aj(t)

xk(t) .

This means that at the new time step each agent places itself in the barycenter ofthe position of all agents which it sees in a neighborhood of radius R around it.


Krause’s model - II

Equivalently, one has

xj(t+ 1) = xj(t) + uj(t) = xj(t) +∑k∈A

Pjk (xk(t)− xj(t)) ,

with

uj(t) =

1Nj(t)

∑k∈Aj(t)

xk(t)

− xj(t) =1

Nj(t)

∑k∈Aj(t)

(xk(t)− xj(t)) ,

i.e., each agent sets its velocity to the difference between the barycenter indicatedabove and its current position.

Thus, the communication matrix P = P (t) has elements given by

Pjk =

{Nj(t)

−1 if |xk(t)− xj(t)| ≤ R ,

0 otherwise .


Krause’s model - II

Equivalently, one has

xj(t+ 1) = xj(t) + uj(t) = xj(t) +∑k∈A


with

uj(t) =

1Nj(t)

∑k∈Aj(t)

xk(t)

− xj(t) =1

Nj(t)

∑k∈Aj(t)

(xk(t)− xj(t)) ,

i.e., each agent sets its velocity to the difference between the barycenter indicatedabove and its current position.

Thus, the communication matrix P = P (t) has elements given by

Pjk =

{Nj(t)

−1 if |xk(t)− xj(t)| ≤ R ,

0 otherwise .


A general consensus model

In a more general setting, the dynamics we are going to consider is described bythe law

xj(t+ 1) = xj(t) +∑k∈A


with

Pjk =1N

Φ(xk(t)− xj(t)) ,

where Φ : Rq → [0, 1] is a function such that Φ(x) = Φ(−x) for all x.

For Krause’s model, we have Φ = χB(0,R).

Property: The barycenter of the system is preserved by such a dynamics.




xj(t+ 1) = xj(t) +∑k∈A


with

Pjk =1N








xj(t+ 1) = xj(t) +∑k∈A


with

Pjk =1N






Extensions - I

Stochastic gossip models(e.g., Deffuant and Weisbuch (2000), Como and Fagnani (2010),... )describe the opinion evolution generated by pairwise random interactions.

Each agent j is activated at the ticking of an independent rate-1 Poisson clock,and randomly choses one other agent, say k.

As a result, the opinion of j (and, possibly, of k) is updated to a convexcombination of their current opinions, depending upon a confidence estimation:

if Xj(t) = x and Xk(t) = y, then

Xj(t+ 1) = x+ Φ(y − x)(y − x) .


Extensions - II

Another relevant field of investigation concerns modeling the movement ofpopulation of animals (flocking, or herding, or schooling). See, e.g.,

Tsitsiklis (1984), Vicsek et al (1995), Jadbabaie, Lin and Morse (2003), Cuckerand Smale (2007), ...

In particular, Cucker and Smale propose the following second-order model:

xj(t+ 1) = xj(t) + vj(t) ,

vj(t+ 1) = vj(t) +∑k∈A

Pjk(vk(t)− vj(t)

),

where the elements of the communication matrix P are given by

Pjk =H

(1 + ‖xk(t)− xj(t)‖2)β

for some fixed H > 0 and β ≥ 0.


Extensions - II

Another relevant field of investigation concerns modeling the movement ofpopulation of animals (flocking, or herding, or schooling). See, e.g.,

Tsitsiklis (1984), Vicsek et al (1995), Jadbabaie, Lin and Morse (2003), Cuckerand Smale (2007), ...

In particular, Cucker and Smale propose the following second-order model:

xj(t+ 1) = xj(t) + vj(t) ,

vj(t+ 1) = vj(t) +∑k∈A

Pjk(vk(t)− vj(t)

),

where the elements of the communication matrix P are given by

Pjk =H

(1 + ‖xk(t)− xj(t)‖2)β

for some fixed H > 0 and β ≥ 0.


The continuous-in-space model

When the number of agents N is very large, one can identify the set of agents attime t with a mass distribution µt in Rq.

Since agents are neither created nor destroyed, the total mass of µt is preserved,hence (up to a normalization) it is not restrictive to assume that µt is, at everytime t ≥ 0, a probability measure in Rq.

In principle, µt can be any Borel probability measure in Rq,such as a (normalized) Lebesgue measure in [a, b], or a fully atomic measure

µt = 1N

∑Nj=1 δxj(t).

A velocity field Vt(x) = Vt(µt)(x) is attached to any point x ∈ Rq at time t.

Lagrangean approach −→ Eulerian approach






µt = 1N

∑Nj=1 δxj(t).








µt = 1N

∑Nj=1 δxj(t).








µt = 1N

∑Nj=1 δxj(t).




The “push-forward”

In a discrete-in-time setting, the dynamical system takes the form

µt+1 = T (µt)µt = γt#µt, t = 0, 1, 2, . . . ,

whereT (µt) = γt# ·

is the push-forward of a measure by the mapping

γt : suppµt ⊆ Rq → Rq, γt(x) = x+ Vt(µt)(x) ,

which is formally defined as

γt#µt(E) = µt(γ−1t (E)) for every Borel set E.


A mass transportation problem

Equivalently, we have∫Rq

ϕ(x) dµt+1 =∫

Rq

ϕ(x+ Vt(x)) dµt

for every (bounded and Borel) function ϕ.

By choosing ϕ as the characteristic function of a set E one may substantiate theintuitive idea that a point x in the support of µt moves at time t+ 1 to the pointγt(x) = x+ Vt(x) in the support of µt+1.

This formulation is an instance of mass transportation problem.




ϕ(x) dµt+1 =∫

Rq

ϕ(x+ Vt(x)) dµt







ϕ(x) dµt+1 =∫

Rq

ϕ(x+ Vt(x)) dµt





The Eulerian velocity

The velocity field is defined, for any x ∈ supp(µt), as

Vt(x) = Vt(µt)(x) =∫

Rq

Φ(y − x) y dµt(y)− x =∫

Rq

Φ(y − x) (y − x) dµt(y) ,

where again Φ : Rq → [0, 1] is a measurable functions satisfying Φ(x) = Φ(−x)for all x.

Note the non-local dependence of Vt(x) upon x.


The Eulerian velocity

The velocity field is defined, for any x ∈ supp(µt), as

Vt(x) = Vt(µt)(x) =∫

Rq

Φ(y − x) y dµt(y)− x =∫

Rq

Φ(y − x) (y − x) dµt(y) ,

where again Φ : Rq → [0, 1] is a measurable functions satisfying Φ(x) = Φ(−x)for all x.

Note the non-local dependence of Vt(x) upon x.


The continuous-in time model

The continuous-in-time counterpart is the conservation law (continuity equation)

∂

∂tµt + div Vtµt = 0 ,

to be meant in the sense of measures, i.e.,

d

dt

∫Rq

η(x) dµt(x) =∫

Rq

∇η(x) · Vt(x) dµt(x)

for any test function η ∈ D(Rq).

We say that a family of probability measures µt, t ≥ 0, is a solution, if for everytest function η(x) ∈ D(Rq), the function

t 7→∫

Rq

η(x) dµt(x), t ≥ 0,

is continuous in [0,∞), differentiable in (0,∞) and satisfies the previous equationfor every t > 0.




∂



d

dt

∫Rq

η(x) dµt(x) =∫

Rq




t 7→∫

Rq

η(x) dµt(x), t ≥ 0,





∂



d

dt

∫Rq

η(x) dµt(x) =∫

Rq




t 7→∫

Rq

η(x) dµt(x), t ≥ 0,



The case of absolutely continuous measures

Assume that the probability measures µt are absolutely continuous with respect tothe Lebesgue measure on Rq, i.e., there exists a density function ρ(t, x) ≥ 0, whichfor all t is compactly supported in x and satisfies

∫Rq ρ(t, x) dx = 1, such that

dµt = ρ(t, x) dx .

Then, the continuity equation becomes

∂ρ

∂t+ divF = 0 ,

where the flux F is the nonlocal function

F (ρ; t, x) = V (ρ; t, x)ρ(t, x)

depending on the velocity field

V (ρ; t, x) =∫

Rq

Φ(y − x) (y − x) ρ(t, y) dy .







∂ρ

∂t+ divF = 0 ,


F (ρ; t, x) = V (ρ; t, x)ρ(t, x)


V (ρ; t, x) =∫

Rq

Φ(y − x) (y − x) ρ(t, y) dy .







∂ρ

∂t+ divF = 0 ,


F (ρ; t, x) = V (ρ; t, x)ρ(t, x)


V (ρ; t, x) =∫

Rq

Φ(y − x) (y − x) ρ(t, y) dy .


Existence and Uniqueness

Assume that V is given by Vt(x) =∫

Rq Φ(y − x) (y − x) dµt(y), whereΦ : Rq → [0, 1] is a measurable functions satisfying Φ(x) = Φ(−x) for all x.

TheoremLet µ0 be any probability measure on Rq with compact support. Then

the discrete-in-time dynamical system

µt+1 = T (µt)µt , t = 0, 1, 2, . . . ,

generates a sequence of probability measures which converge, as t→∞, to alimit probability measure µ∞ in any Wasserstein p-distance.

If there exist R > 0 and δ > 0 such that Φ(x) > 0 whenever |x| < R, thenµ∞ is a purely atomic measure, whose atoms are a distance at least R apartfrom one another.

A similar result holds for the continuous-in-time dynamical system, with theadditional result of the uniqueness of the solution.


1D representative results

The evolution of a piecewise constant density towards steady state:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50

0

50

100

150

200

250

300


Constant initial density

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5

2

2.5

3

3.5

4

4.5

Position of deltas (horizontal axis) vs | logR| (vertical axis)


Linear density evolution

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

0

10

20

30

40

50

60

initial asymptotic


Parabolic density evolution

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

0

10

20

30

40

50

60

70

80

initial asymptotic


Linear initial density

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5

2

2.5

3

3.5

4

Position of deltas (horizontal axis) vs | logR| (vertical axis)


Numerics: a “push-forward” algorithm

Fix any partition D of Rq made of mutually disjoint, bounded Borel sets D. Forany E ∈ D, we have

µt+1(E) = µt(γ−1t (E)

)=∑D∈D

µt(D ∩ γ−1

t (E)).

Now, we make the following assumptions:

i) In each D ∈ D, µt is approximated by a multiple of the Lebesgue measuretherein, i.e.,

µt|D ∼ ρDt dx|D , with ρDt = µt(D)/|D| .

ii) In each D ∈ D, the velocity Vt is approximated by a constant velocity V Dt , sothat γt is approximated therein by the affine invertible mapping

γDt (x) = x+ τV Dj .

Note that such a γDt is a rigid motion, so that |X| = |γDt (X)| for anymeasurable X ⊆ D.




µt+1(E) = µt(γ−1t (E)

)=∑D∈D

µt(D ∩ γ−1

t (E)).










µt+1(E) = µt(γ−1t (E)

)=∑D∈D

µt(D ∩ γ−1

t (E)).










µt+1(E) = µt(γ−1t (E)

)=∑D∈D

µt(D ∩ γ−1

t (E)).










µt+1(E) = µt(γ−1t (E)

)=∑D∈D

µt(D ∩ γ−1

t (E)).








A “push-forward” algorithm (cont’d)

D

γDt (D)

E

γDt

Pictorial respresentation of the “push-forward” algorithm

Using these approximations, we obtain

µt+1(E) = ρEt+1|E| ∼∑D∈D

ρDt |D ∩(γDt)−1

(E)| =∑D∈D

ρDt |γDt (D) ∩ E| ,

which provides the following approximate dynamics for the local mass densities:

ρEt+1 ∼∑D∈D

|γDt (D) ∩ E||E|

ρDt , ∀E ∈ D .



D

γDt (D)

E

γDt



µt+1(E) = ρEt+1|E| ∼∑D∈D


(E)| =∑D∈D



ρEt+1 ∼∑D∈D

|γDt (D) ∩ E||E|

ρDt , ∀E ∈ D .



D

γDt (D)

E

γDt



µt+1(E) = ρEt+1|E| ∼∑D∈D


(E)| =∑D∈D



ρEt+1 ∼∑D∈D

|γDt (D) ∩ E||E|

ρDt , ∀E ∈ D .


The numerical algorithm

The actual numerical algorithm is defined by fixing a stepsize h > 0 and choosingthe partition of Rq given by the cells

Dj = xj + h[−1/2, 1/2]q

centered at the lattice points xj = jh, with j = (j1, . . . , jq) ∈ Zq.

Writing ρjt for ρDj

t etc..., the approximate velocity in the cell Dj is defined as

V jt = Vt(xj) =∑k∈Zq

∫Dk

Φ(y − xj) (y − xj)ρkt dx .

With these definitions, we consider the following algorithm:

ρjt+1 =∑k∈Zq

|γkt (Dk) ∩Dj ||Dj |

ρkt , ∀j ∈ Zq .




Dj = xj + h[−1/2, 1/2]q





∫Dk



ρjt+1 =∑k∈Zq






Dj = xj + h[−1/2, 1/2]q





∫Dk



ρjt+1 =∑k∈Zq




The numerical algorithm (cont’d)

We assume that the time step τ = τt is chosen in such a way that the Courant-Friedrichs-Lewy (CFL) condition

τth

maxk∈Zq

|V kt |∞ = 1

is fulfilled for each t.

Then, the algorithm simplifies as

ρjt+1 =∑

k : ‖j−k‖∞≤1




The numerical algorithm (cont’d)

We assume that the time step τ = τt is chosen in such a way that the Courant-Friedrichs-Lewy (CFL) condition

τth

maxk∈Zq

|V kt |∞ = 1

is fulfilled for each t.

Then, the algorithm simplifies as

ρjt+1 =∑

k : ‖j−k‖∞≤1




Convergence of the numerical algorithm

The proposed scheme can be viewed as a genuinely multidimensional Upwindscheme for discretizing the conservation law

∂

∂tµt + div Vtµt = 0

in the space of probabilistic measures.

TheoremFor any h, convergence as t→∞ of the discrete measures µt,h to a limitdiscrete measure µ∞,h occurs in any Wasserstein p-distance.

The limit measure is a finite sum of “discrete deltas”, at a distance at leastR apart from one another.


Convergence of the numerical algorithm

The proposed scheme can be viewed as a genuinely multidimensional Upwindscheme for discretizing the conservation law

∂

∂tµt + div Vtµt = 0

in the space of probabilistic measures.

TheoremFor any h, convergence as t→∞ of the discrete measures µt,h to a limitdiscrete measure µ∞,h occurs in any Wasserstein p-distance.

The limit measure is a finite sum of “discrete deltas”, at a distance at leastR apart from one another.


Structure of the discrete limit densities

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

One can prove that, as t→∞, the support of any limit “discrete delta” consistsof a patch of 2q neighboring cell. Simmetry may further reduce the support.


2D numerical results

Position and strength of deltas, starting from a uniform density in a square:(top to bottom, left to right: R = 0.30, R = 0.23, R = 0.15 and R = 0.10)


2D results - the square case (cont’d)

Table: The square case, for R = 0.15: concentrated measures aδ at shown locations.Approximate values of the constants a vs h

1/h © ∆ �32 0.040 0.079 0.16264 0.064 0.092 0.142

128 0.093 0.103 0.122256 0.095 0.106 0.120


2D results - another case

Position and strength of deltas, starting from a uniform density in a horseshoe:(top to bottom, left to right: R = 0.40, R = 0.30, R = 0.15 and R = 0.08)


Invariance and stability

A probability measure µ is said to have a radial symmetry with respect tox0 ∈ Rq if for any rotation U centered at x0 one has U#µ = µ.

TheoremIn dimension q > 1, let Φ be a radial function. If the initial measure µ0 has radialsymmetry with respect to some x0, then µ∞ = δx0 .

Thus, in 2D, if µ0 is the characteristic function of the unit circle, then µ∞ = δ0.

However, ....


Invariance and stability (cont’d)

0

0.5

1

1.5

2

0

0.5

1

1.5

2

−10

0

10

20

30

40

50

60

70

80

90

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Rotational invariance is unstable.


Conclusions and further developments

An Eulerian approach has been proposed to study consensus models

Theoretical and numerical investigation has shown that the Rendez-vousProblem can be succesfully solved only if the communication length issufficiently large

A sort of bifurcation diagram for the limit patterns has been obtained

The proposed numerical scheme stems from a faithful realization of the“push-forward” dynamics in the space of probability measures

Developments include

Enhance efficiency by time-step adaptivity, and space localization (for largedimensions q)

Consider heterogenous models, where agents are split into families, each onewith a different attitude to interact

Extend the push-forward algorithm to simulate the random gossip model

Incorporate further randomness meant as noise in the whole system (opinionsmay also change when no interaction takes place)

...


Conclusions and further developments

An Eulerian approach has been proposed to study consensus models

Theoretical and numerical investigation has shown that the Rendez-vousProblem can be succesfully solved only if the communication length issufficiently large

A sort of bifurcation diagram for the limit patterns has been obtained

The proposed numerical scheme stems from a faithful realization of the“push-forward” dynamics in the space of probability measures

Developments include

Enhance efficiency by time-step adaptivity, and space localization (for largedimensions q)

Consider heterogenous models, where agents are split into families, each onewith a different attitude to interact

Extend the push-forward algorithm to simulate the random gossip model

Incorporate further randomness meant as noise in the whole system (opinionsmay also change when no interaction takes place)

...Claudio Canuto () Consensus models and their simulation 37 / 39

References (by the authors)

C. Canuto, F. Fagnani, P. Tilli, “A Eulerian approach to the analysis ofrendez-vous algorithms”, Proceedings of IFAC2008, Seoul, Korea, July 6-11,pp. 9039-9044, 2008.

C. Canuto, F. Fagnani, P. Tilli, “An Eulerian approach to the analysis ofKrause’s consensus models”, submitted 2010.http://calvino.polito.it/ fagnani/coordincontrol/euler.pdf

G. Como, and F. Fagnani, “Scaling limits for continuous opinion dynamicssystems”, to appear in Annals of Applied Probabilityhttp://calvino.polito.it/ fagnani/coordincontrol/scaling.pdf

F. Fagnani, S. Zampieri, “Randomized consensus algorithms over large scalenetworks”, IEEE J. on Selected Areas of Communications, vol. 26, pp.634-649, 2008.


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