## Section: Research Program

### Local Interactions and Transient Analysis in Adaptive Dynamic Systems

Participants : Jonatha Anselmi, Nicolas Gast, Bruno Gaujal, Florence Perronnin, Jean-Marc Vincent, Panayotis Mertikopoulos.

Many systems can be effectively described by stochastic population
models. These systems are composed of a set of $n$ entities
interacting together and the resulting stochastic process can be
seen as a continuous-time Markov chain with a finite state
space. Many numerical techniques exist to study the behavior of
Markov chains, to solve stochastic optimal control
problems [93] or to perform
model-checking [52]. These techniques, however, are
limited in their applicability, as they suffer from the *curse
of dimensionality*: the state-space grows exponentially with $n$.

This results in the need for approximation techniques. Mean field analysis offers a viable, and often very accurate, solution for large $n$. The basic idea of the mean field approximation is to count the number of entities that are in a given state. Hence, the fluctuations due to stochasticity become negligible as the number of entities grows. For large $n$, the system becomes essentially deterministic. This approximation has been originally developed in statistical mechanics for vary large systems composed of more than ${10}^{20}$ particles (called entities here). More recently, it has been claimed that, under some conditions, this approximation can be successfully used for stochastic systems composed of a few tens of entities. The claim is supported by various convergence results [74], [83], [106], and has been successfully applied in various domains: wireless networks [56], computer-based systems [77], [88], [101], epidemic or rumour propagation [66], [81] and bike-sharing systems [70]. It is also used to develop distributed control strategies [105], [87] or to construct approximate solutions of stochastic model checking problems [58], [59], [60].

Within the POLARIS project, we will continue developing both the theory behind these approximation techniques and their applications. Typically, these techniques require a homogeneous population of objects where the dynamics of the entities depend only on their state (the state space of each object must not scale with $n$ the number of objects) but neither on their identity nor on their spatial location. Continuing our work in [74], we would like to be able to handle heterogeneous or uncertain dynamics. Typical applications are caching mechanisms [77] or bike-sharing systems [71]. A second point of interest is the use of mean field or large deviation asymptotics to compute the time between two regimes [96] or to reach an equilibrium state. Last, mean-field methods are mostly descriptive and are used to analyse the performance of a given system. We wish to extend their use to solve optimal control problems. In particular, we would like to implement numerical algorithms that use the framework that we developed in [75] to build distributed control algorithms [68] and optimal pricing mechanisms [76].