In the first part of this thesis I consider site and bond percolation on a Random Connection Model and prove that for a wide range of connection functions the critical site probability is strictly greater than the critical bond probability and use this fact to improve previously known non-strict inequalities to strict inequalities. In the second part I consider percolation on the even phase of a Random Sequential Adsorption model and prove that the critical intensity is finite and strictly bigger than 1. Both of these main results make use of an enhancement technique.

We formulate a novel approach to infer conditional independence models or Markov structure of a multivariate distribution. Specifically, our objective is to place informative prior distributions over graphs (decomposable and unrestricted) and sample efficiently from the induced posterior distribution. We also explore the idea of factorizing according to complete sets of a graph; which implies working with a hypergraph that cannot be retrieved from the graph alone. The key idea we develop in this paper is a parametrization of hypergraphs using the geometry of points in $R^m$. This induces informative priors on graphs from specified priors on finite sets of points. Constructing hypergraphs from finite point sets has been well studied in the fields of computational topology and random geometric graphs. We develop the framework underlying this idea and illustrate its efficacy using simulations. / Dissertation

This dissertation has its focus on two different topics: 1. non-asymptotic analysis of polar codes as a new paradigm in error correcting codes with very promising features, and 2. network properties for wireless networks of practical size. In its first part, we investigate properties of polar codes that can be potentially useful in real-world applications. We start with analyzing the performance of finite-length polar codes over the binary erasure channel (BEC), while assuming belief propagation (BP) as the decoding method. We provide a stopping set analysis for the factor graph of polar codes, where we find the size of the minimum stopping set. Our analysis along with bit error rate (BER) simulations demonstrates that finite-length polar codes show superior error floor performance compared to the conventional capacity-approaching coding techniques. Motivated by good error floor performance, we introduce a modified version of BP decoding while employing a guessing algorithm to improve the BER performance. Each application may impose its own requirements on the code design. To be able to take full advantage of polar codes in practice, a fundamental question is which practical requirements are best served by polar codes. For example, we will see that polar codes are inherently well-suited for rate-compatible applications and they can provably achieve the capacity of time-varying channels with a simple rate-compatible design. This is in contrast to LDPC codes for which no provably universally capacity-achieving design is known except for the case of the erasure channel. This dissertation investigates different approaches to applications such as UEP, rate-compatible coding, and code design over parallel sub-channels (non-uniform error correction). Furthermore, we consider the idea of combining polar codes with other coding schemes, in order to take advantage of polar codes' best properties while avoiding their shortcomings. Particularly, we propose, and then analyze, a polar code-based concatenated scheme to be used in Optical Transport Networks (OTNs) as a potential real-world application The second part of the dissertation is devoted to the analysis of finite wireless networks as a fundamental problem in the area of wireless networking. We refer to networks as being finite when the number of nodes is less than a few hundred. Today, due to the vast amount of literature on large-scale wireless networks, we have a fair understanding of the asymptotic behavior of such networks. However, in real world we have to face finite networks for which the asymptotic results cease to be valid. Here we study a model of wireless networks, represented by random geometric graphs. In order to address a wide class of the network's properties, we study the threshold phenomena. Being extensively studied in the asymptotic case, the threshold phenomena occurs when a graph theoretic property (such as connectivity) of the network experiences rapid changes over a specific interval of the underlying parameter. Here, we find an upper bound for the threshold width of finite line networks represented by random geometric graphs. These bounds hold for all monotone properties of such networks. We then turn our attention to an important non-monotone characteristic of line networks which is the Medium Access (MAC) layer capacity, defined as the maximum number of possible concurrent transmissions. Towards this goal, we provide a linear time algorithm which finds a maximal set of concurrent non-interfering transmissions and further derive lower and upper bounds for the cardinality of the set. Using simulations, we show that these bounds serve as reasonable estimates for the actual value of the MAC-layer capacity.

Dans cette thèse, on s'intéresse à des algorithmes d'analyse de données utilisant des marches aléatoires sur des graphes de voisinage, ou graphes géométriques aléatoires, construits à partir des données. On sait que les marches aléatoires sur ces graphes sont des approximations d'objets continus appelés processus de diffusion. Dans un premier temps, nous utilisons ce résultat pour proposer un nouvel algorithme de partitionnement de données flou de type recherche de modes. Dans cet algorithme, on définit les paquets en utilisant les propriétés d'un certain processus de diffusion que l'on approche par une marche aléatoire sur un graphe de voisinage. Après avoir prouvé la convergence de notre algorithme, nous étudions ses performances empiriques sur plusieurs jeux de données. Nous nous intéressons ensuite à la convergence des mesures stationnaires des marches aléatoires sur des graphes géométriques aléatoires vers la mesure stationnaire du processus de diffusion limite. En utilisant une approche basée sur la méthode de Stein, nous arrivons à quantifier cette convergence. Notre résultat s'applique en fait dans un cadre plus général que les marches aléatoires sur les graphes de voisinage et nous l'utilisons pour prouver d'autres résultats : par exemple, nous arrivons à obtenir des vitesses de convergence pour le théorème central limite. Dans la dernière partie de cette thèse, nous utilisons un concept de topologie algébrique appelé homologie persistante afin d'améliorer l'étape de "pooling" dans l'approche "sac-de-mots" pour la reconnaissance de formes 3D. / In this thesis, we study data analysis algorithms using random walks on neighborhood graphs, or random geometric graphs. It is known random walks on such graphs approximate continuous objects called diffusion processes. In the first part of this thesis, we use this approximation result to propose a new soft clustering algorithm based on the mode seeking framework. For our algorithm, we want to define clusters using the properties of a diffusion process. Since we do not have access to this continuous process, our algorithm uses a random walk on a random geometric graph instead. After proving the consistency of our algorithm, we evaluate its efficiency on both real and synthetic data. We then deal tackle the issue of the convergence of invariant measures of random walks on random geometric graphs. As these random walks converge to a diffusion process, we can expect their invariant measures to converge to the invariant measure of this diffusion process. Using an approach based on Stein's method, we manage to obtain quantitfy this convergence. Moreover, the method we use is more general and can be used to obtain other results such as convergence rates for the Central Limit Theorem. In the last part of this thesis, we use the concept of persistent homology, a concept of algebraic topology, to improve the pooling step of the bag-of-words approach for 3D shapes.