"Sparsity and Clustering in multi-agent systems"

Résumé :
This manuscript is partitioned in three parts. The first one is written in French and contains an extended resume. I start by briefly presenting my main teaching activities as well as my main contributions to the formation proposed at ENSEM. An important part summarizes my research activity starting with the contributions of my Ph.D. thesis and finishing with the last results that I obtained at CRAN. The presentation includes the list of Ph.D. and Master students that I supervised. This part also contains complete lists of grants and publications in which I was involved. For the sake of accessibility, the last two parts of the manuscript are written in English. The second part of this work provides a selection of results that I obtained on the topic of multi-agent systems. The basis of this research is the fact that real large-scale networks are partitioned in several clusters. The first question that arises in this context is related to the cluster detection in a decentralized way. It is also natural to analyze these networks from the perspective of continuous interactions inside clusters and sporadic-discrete ones outside. The first chapter is dedicated to the description of the basic concepts and the main results existing in the consensus literature. Chapter 2 presents a class of discrete time multi-agent systems modeling opinion dynamics with decaying confidence. This opinion dynamics model is applied to address the problem of community detection in graphs. Basically the convergence speed is higher inside clusters and the decaying confidence allows to cut the links between different clusters. Chapter 3 addresses the problem of coordination in heterogeneous networks containing both linear and linear impulsive agents. Precisely we assume that several clusters/communities exist in the network. The agents inside one cluster can continuously interact with each others. On top of this, few agents in each cluster can interact at some discrete time instants outside their own cluster. We finish the presentation with one part containing some of my research perspectives. A first research direction of my future works is at the intersection of singular perturbation theory and multi-agent systems' analysis. Two different aspects will be addressed. The first one is directly related to the results presented in Part 2. Precisely, the presence of clusters in the network generates, as pointed out before, a time-scale separation. In other words, the agents will agree faster inside a cluster than in the whole network. The second aspect considers the consensus problem in networks of singularly perturbed agents e.g. whose state components evolve on different time-scales. A second research direction focusses on the analysis of opinion dynamics in social networks. We propose a consensus like model based on quantized information about the neighbors opinion. The main advantages of this model are that it captures oscillatory behaviors of opinions as well as the dissensus in the network.