A significant progress in the evolution of the computer systems and their interconnection over the past 70 years has allowed replacing the frequently used centralized architectures with the highly distributed ones, formed by independent entities fulfilling specific functionalities as one user-intransparent unit. This has resulted in an intense scientic interest in distributed algorithms and their frequent implementation into real systems. Especially, distributed algorithms for multi-sensor data fusion, ensuring an enhanced QoS of executed applications, find a wide usage. This doctoral thesis addresses an optimization and an analysis of the distributed systems, namely the distributed consensus-based algorithms for an aggregate function estimation (primarily, my attention is focused on a mean estimation). The first section is concerned with a theoretical background of the distributed systems, their evolution, their architectures, and a comparison with the centralized systems (i.e. their advantages/disadvantages). The second chapter deals with multi-sensor data fusion, its application, the classification of the distributed estimation techniques, their mathematical modeling, and frequently quoted algorithms for distributed averaging (e.g. protocol Push-Sum, Metropolis-Hastings weights, Best Constant weights etc.). The practical part is focused on mechanisms for an optimization of the distributed systems, the proposal of novel algorithms and complements for the distributed systems, their analysis, and comparative studies in terms of such as the convergence rate, the estimation precision, the robustness, the applicability to real systems etc.

Les protocoles de consensus ont gagné beaucoup d’intérêt ces dernières années. Dans cette thèse, nous étudions les problèmes d’optimisation, de contrôle, et de théorie de jeu qui se posent dans ces protocoles. Tout d’abord, nous étudions les techniques d’optimisation pour des problèmes de sélection de poids permettant ainsi d’augmenter la vitesse de convergence de protocoles de consensus dans les réseaux. Nous proposons de sélectionner les poids en appliquant un algorithme d’approximation: minimisation de la norme p de Schatten de la matrice de poids. Nous caractérisons l’erreur induite par cette approximation et nous montrons que l’algorithme proposé a l’avantage qu’il peut être soit résolu de façon distribuée. Ensuite, nous proposons un cadre conceptuel d’analyse des jeux d’adversaire qui peut ajouter du bruit aux poids utilisés par l’algorithme de consensus de moyenne afin d’éloigner le système de consensus. Nous analysons également la performance des algorithmes de consensus de moyenne où les informations échangées entre les agents voisins sont soumises à la quantification uniforme déterministe (les valeurs réelles envoyées par les nœuds de leurs voisins sont tronquées). Le problème de la terminaison des protocoles de consensus s’avère difficile dans le cadre distribué. Nous proposons un algorithme distribué pour la terminaison des protocoles de consensus. L’algorithme réduit la charge de communication tout en garantissant la convergence vers un consensus. Enfin, nous proposons une mesure de similarité qui évalue la qualité d’un regroupement (clustering) des nœuds dans un réseau. Un algorithme local de clustering basé sur cette métrique est donné. / Consensus protocols have gained a lot of interest in the recent years. In this thesis, we study optimization, control, and game theoretical problems arising in consensus protocols. First, we study optimization techniques for weight selection problems to increase the speed of convergence of discrete-time consensus protocols on networks. We propose to select the weights by applying an approximation algorithm: minimizing the Schatten p-norm of the weight matrix. We characterize the approximation error and we show that the proposed algorithm has the advantage that it can be solved in a totally distributed way. Then we propose a game theoretical framework for an adversary that can add noise to the weights used by averaging protocols to drive the system away from consensus. We give the optimal strategies for the game players (the adversary and the network designer) and we show that a saddle-point equilibrium exists in mixed strategies. We also analyze the performance of distributed averaging algorithms where the information exchanged between neighboring agents is subject to deterministic uniform quantization (e.g., when real values sent by nodes to their neighbors are truncated). Consensus algorithms require that nodes exchange messages persistently to reach asymptotically consensus. We propose a distributed algorithm that reduces the communication overhead while still guaranteeing convergence to consensus. Finally, we propose a score metric that evaluates the quality of clusters such that the faster the random walk mixes in the cluster and the slower it escapes, the higher is the score. A local clustering algorithm based on this metric is proposed.