How much is it possible to perturb a mean-field interaction ?

A convenient framework for the study of synchronization of interacting individuals is to consider systems of exchangeable particles interacting on the complete graph. This assumption allows for a rigorous analysis of the behavior of the system in the limit of large population, at least on a bounded time interval. From a practical point of view (e.g. simulations), a crucial question is how to derive from this formalism the behavior of the system on a longer time scale (that is dependent on the size of the population). I will first discuss this issue on the particular case of the Kuramoto model. And what if the graph of interaction is no longer the complete graph? Informally, in a situation where each particle has "enough" neighbors to interact with, it is natural to expect a similar mean-field behavior. The second point of the talk will be to show that this statement holds for a large class of (possibly inhomogeneous and sparse) random graphs of interaction, at least on a bounded time interval.

This is joint works with Christophe Poquet (Lyon 1) and Giambattista Giacomin and Sylvain Delattre (Paris 7).