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, , of `agents' evolving according to
quite simple dynamical systems coupled in according to some `locality'
constraint. Each agent maintains a time function
representing the `opinion', the `belief' it has on something.
As time elapses, agent 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 from its
current opinion.
The main interest for these models is for quite large.
Mathematically, this means that one takes the limit for .
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 also gives rise to a natural numerical
algorithm based on the `push forward' scheme, which
allows one to perform numerical simulations with complexity
independent on the number of agents, and in a genuinely
multi-dimensional manner.

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