# Multi-agent cooperative dynamical systems: Theory and numerical<br/><br/>simulations

Monday, November 1, 2010 - 12:00pm - 12:45pm

Keller 3-180

Claudio Canuto (Politecnico di Torino)

We are witnessing an increasing interest for cooperative

dynamical systems proposed in the recent literature as possible

models for

opinion dynamics in social and economic networks.

Mathematically,

they consist of a large number, N, of 'agents' evolving

according to

quite simple dynamical systems coupled in according to some

'locality'

constraint. Each agent i maintains a time function

x

representing the 'opinion,' the 'belief' it has on something.

As time elapses, agent i interacts with neighbor agents

and modifies its opinion by averaging it with

the one of its neighbors. A critical issue is the way

'locality'

is modelled and interaction takes place. In Krause's model

each agent can see the opinion of all the others but

averages with only those which are within a threshold R from

its current opinion.

The main interest for these models is for N quite large.

Mathematically, this means that one takes the limit for N →

+ ∞.

We adopt an Eulerian approach, moving focus from opinions of

various

agents to distributions of opinions. This leads to a sort of

master

equation which is a PDE in the space of probabily measures; it

can be

analyzed by the techniques of Transportation Theory, which

extends

in a very powerful way the Theory of Conservation Laws.

Our Eulerian approach gives rise to a natural numerical

algorithm based on the `push forward' of measures, which

allows one to perform numerical simulations with complexity

independent on the number of agents, and in a genuinely

multi-dimensional manner.

We prove the existence of a limit measure as t → ∞,

which

for the exact dynamics is purely atomic with atoms at least at

distance R apart,

whereas for the numerical dynamics it is 'almost purely atomic'

(in

a precise sense). Several representative examples will be

discussed.

This is a joint work with Fabio Fagnani and Paolo Tilli.

dynamical systems proposed in the recent literature as possible

models for

opinion dynamics in social and economic networks.

Mathematically,

they consist of a large number, N, of 'agents' evolving

according to

quite simple dynamical systems coupled in according to some

'locality'

constraint. Each agent i maintains a time function

x

_{i}(t)representing the 'opinion,' the 'belief' it has on something.

As time elapses, agent i interacts with neighbor agents

and modifies its opinion by averaging it with

the one of its neighbors. A critical issue is the way

'locality'

is modelled and interaction takes place. In Krause's model

each agent can see the opinion of all the others but

averages with only those which are within a threshold R from

its current opinion.

The main interest for these models is for N quite large.

Mathematically, this means that one takes the limit for N →

+ ∞.

We adopt an Eulerian approach, moving focus from opinions of

various

agents to distributions of opinions. This leads to a sort of

master

equation which is a PDE in the space of probabily measures; it

can be

analyzed by the techniques of Transportation Theory, which

extends

in a very powerful way the Theory of Conservation Laws.

Our Eulerian approach gives rise to a natural numerical

algorithm based on the `push forward' of measures, which

allows one to perform numerical simulations with complexity

independent on the number of agents, and in a genuinely

multi-dimensional manner.

We prove the existence of a limit measure as t → ∞,

which

for the exact dynamics is purely atomic with atoms at least at

distance R apart,

whereas for the numerical dynamics it is 'almost purely atomic'

(in

a precise sense). Several representative examples will be

discussed.

This is a joint work with Fabio Fagnani and Paolo Tilli.

MSC Code:

70G60

Keywords: