We study the existence of expander graphs with a focus on odd and unique expanders. The main goal is to describe configurations of arguments for which there is no infinite family of expanders. The most imporant result is that for every graph there is a nonempty subset of at most half of its vertices, such that every other vertex is connected at least twice to the subset or not connected to the subset at all. It follows that certain classes of unique expanders cannot exist. On the other hand we present some configurations for which there are families of expanders. Powered by TCPDF (www.tcpdf.org)

The subject of this thesis is searching in unstructured peer-to-peer systems. Such systems have been used for a variety of different applications, including file-sharing, content distribution and video streaming. These applications have been very popular; they contribute to a large percentage of today's Internet traffic and their users typically number in the millions. By searching, we refer to the process of locating content stored by peers. Searching in unstructured peer-to-peer systems poses a challenge because of high churn: both the topology and the content stored by peers can change quickly as peers arrive and depart, while the network formed under this churn process can be arbitrary at any point in time. As a result, a search mechanism must operate without any a priori assumptions on this dynamic topology. Ideally, a search mechanism should be scalable: as, typically, peers have limited bandwidth, the traffic generated by queries should not grow significantly as the peer population increases. Moreover, a search mechanism should also be reliable: if certain content is in the system, searching should locate it with reasonable guarantees. These two goals can be conflicting, as generating more queries increases a mechanism's reliability but decreases its scalability. Hence, a fundamental question regarding searching in unstructured systems is whether a mechanism can exhibit both properties, despite the network's dynamic and arbitrary nature. In this thesis, we show this is indeed the case, by proposing a novel mechanism that is both scalable and reliable. This is shown under a mathematical model that captures the evolution of both network and content in an unstructured system, but is also verified through simulations. To the best of our knowledge, this is the first provably scalable and reliable search mechanism for unstructured peer-to-peer systems. In addition to the above problem, we also consider a hybrid peer-to-peer system, in which the peer-to-peer network co-exists with a central server. The purpose of this hybrid architecture is to reduce the server's traffic by delegating part of it to its clients ---\emph{i.e.}, the peers: a peer wishing to retrieve certain content first propagates a query over the peer-to-peer network, and downloads the content from the server only if the query fails. This hybrid architecture can be used to partially decentralize a content distribution server, a search engine, an online encyclopedia, etc. The trade-off between scalability and reliability translates, in the hybrid case, to a trade-off between the peer and the server traffic loads. We propose a search mechanism under which both loads remain bounded as the peer population grows. This is surprising, and has an important implication: one can construct hybrid peer-to-peer systems that can handle traffic generated by a large (unbounded) peer population, even when both the server and peer bandwidth capacities are limited. Again, this is proved under a model capturing the hybrid system's dynamic nature and verified through simulations. To the best of our knowledge, our work is the first to show that hybrid systems with such properties exist.

We explore the relationships between the computational problem of recognizing expander graphs, and the problem of efficiently approximating proof length in the well-known system of \emph{resolution}. This program builds upon known connections between graph expansion and resolution lower bounds. A proof system $P$ is \emph{(quasi-)automatizable} if there is a search algorithm which finds a $P$-proof of a given formula $f$ in time (quasi)polynomial in the length of a shortest $P$-proof of $f$. It is open whether resolution is (quasi-)automatizable. We prove several conditional non-automatizability results for resolution modulo new conjectures concerning the complexity of identifying bipartite expander graphs. Our reductions use a natural family of formulas and exploit the well-known relationships between expansion and length of resolution proofs. Our hardness assumptions are unsupported; we survey known results as progress towards establishing their plausibility. The major contribution is a conditional hardness result for the quasi-automatizability of resolution.

We explore the relationships between the computational problem of recognizing expander graphs, and the problem of efficiently approximating proof length in the well-known system of \emph{resolution}. This program builds upon known connections between graph expansion and resolution lower bounds. A proof system $P$ is \emph{(quasi-)automatizable} if there is a search algorithm which finds a $P$-proof of a given formula $f$ in time (quasi)polynomial in the length of a shortest $P$-proof of $f$. It is open whether resolution is (quasi-)automatizable. We prove several conditional non-automatizability results for resolution modulo new conjectures concerning the complexity of identifying bipartite expander graphs. Our reductions use a natural family of formulas and exploit the well-known relationships between expansion and length of resolution proofs. Our hardness assumptions are unsupported; we survey known results as progress towards establishing their plausibility. The major contribution is a conditional hardness result for the quasi-automatizability of resolution.

The subject of this thesis is searching in unstructured peer-to-peer systems. Such systems have been used for a variety of different applications, including file-sharing, content distribution and video streaming. These applications have been very popular; they contribute to a large percentage of today's Internet traffic and their users typically number in the millions. By searching, we refer to the process of locating content stored by peers. Searching in unstructured peer-to-peer systems poses a challenge because of high churn: both the topology and the content stored by peers can change quickly as peers arrive and depart, while the network formed under this churn process can be arbitrary at any point in time. As a result, a search mechanism must operate without any a priori assumptions on this dynamic topology. Ideally, a search mechanism should be scalable: as, typically, peers have limited bandwidth, the traffic generated by queries should not grow significantly as the peer population increases. Moreover, a search mechanism should also be reliable: if certain content is in the system, searching should locate it with reasonable guarantees. These two goals can be conflicting, as generating more queries increases a mechanism's reliability but decreases its scalability. Hence, a fundamental question regarding searching in unstructured systems is whether a mechanism can exhibit both properties, despite the network's dynamic and arbitrary nature. In this thesis, we show this is indeed the case, by proposing a novel mechanism that is both scalable and reliable. This is shown under a mathematical model that captures the evolution of both network and content in an unstructured system, but is also verified through simulations. To the best of our knowledge, this is the first provably scalable and reliable search mechanism for unstructured peer-to-peer systems. In addition to the above problem, we also consider a hybrid peer-to-peer system, in which the peer-to-peer network co-exists with a central server. The purpose of this hybrid architecture is to reduce the server's traffic by delegating part of it to its clients ---\emph{i.e.}, the peers: a peer wishing to retrieve certain content first propagates a query over the peer-to-peer network, and downloads the content from the server only if the query fails. This hybrid architecture can be used to partially decentralize a content distribution server, a search engine, an online encyclopedia, etc. The trade-off between scalability and reliability translates, in the hybrid case, to a trade-off between the peer and the server traffic loads. We propose a search mechanism under which both loads remain bounded as the peer population grows. This is surprising, and has an important implication: one can construct hybrid peer-to-peer systems that can handle traffic generated by a large (unbounded) peer population, even when both the server and peer bandwidth capacities are limited. Again, this is proved under a model capturing the hybrid system's dynamic nature and verified through simulations. To the best of our knowledge, our work is the first to show that hybrid systems with such properties exist.

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This thesis presents a series of algorithmic innovations in Combinatorial Compressed Sensing and Persistent Homology. The unifying strategy across these contributions is in translating structural patterns in the underlying data into specific algorithmic designs in order to achieve: better guarantees in computational complexity, the ability to operate on more complex data, highly efficient parallelisations, or any combination of these.

Le système immunitaire adaptatif est capable de produire une réponse spécifique contre presque tous le pathogènes qui agressent notre organisme. Ceci est dû aux anticorps qui sont des protéines secrétées par les cellules B. Les molécules qui provoquent cette réaction sont appelées antigènes : pendant une réponse immunitaire, les cellules B sont soumises à un processus d’apprentissage afin d’améliorer leur capacité à reconnaitre un antigène donne. Ce processus est appelé maturation d’affinité des anticorps. Nous établissons un cadre mathématique très flexible dans lequel nous définissons et étudions des modelés évolutionnaires simplifies inspirés par la maturation d’affinité des anticorps. Nous identifions les éléments constitutifs fondamentaux de ce mécanisme d’évolution extrêmement rapide et efficace : mutation, division et sélection. En commençant par une analyse rigoureuse du mécanisme de mutation dans le Chapitre 2, nous procédons à l’enrichissement progressif du modelé en ajoutant et analysant le processus de division dans le Chapitre 3 ,puis des pressions sélectives dépendantes de l’affinité dans le Chapitre 4. Notre objectif n’est pas de construire un modèle mathématique très détaillé et exhaustif de la maturation d’affinité des anticorps, mais plutôt d’enquêter sur les interactions entre mutation, division et sélection dans un contexte théorique simplifie. On cherche à comprendre comment les différents paramètres biologiques influencent la fonctionnalité du système, ainsi qu’à estimer les temps caractéristiques de l’exploration de l’espace d’états des traits des cellules B. Au-delà des motivations biologiques de la modélisation de la maturation d’affinité des anticorps, l’analyse de ce processus d’apprentissage nous a amenée à concevoir un modèle mathématique qui peut également s’appliquer à d’autres systèmes d’évolution, mais aussi à l’étude de la propagation de rumeurs ou de virus. Notre travail théorique s’accompagne de nombreuses simulations numériques qui viennent soit l’illustrer soit montrer que certains résultats demeurent extensibles a des situations plus compliquées. / The adaptive immune system is able to produce a specific response against almost any pathogen that could penetrate our organism and inflict diseases. This task is assured by the production of antigen-specific antibodies secreted by B-cells. The agents which causes this reaction are called antigens: during an immune response B-cells are submitted to a learning process in order to improve their ability to recognize the immunizing antigen. This process is called antibody affinity maturation. We set a highly flexible mathematical environment in which we define and study simplified mathematical evolutionary models inspired by antibody affinity maturation. We identify the fundamental building blocks of this extremely efficient and rapid evolutionary mechanism: mutation, division and selection. Starting by a rigorous analysis of the mutational mechanism in Chapter 2, we proceed by successively enriching the model by adding and analyzing the division process in Chapter 3 and affinity-dependent selection pressures in Chapter 4. Our aim is not to build a very detailed and comprehensive mathematical model of antibody affinity maturation, but rather to investigate interactions between mutation, division and selection in a simplified theoretical context. We want to understand how the different biological parameters affect the system’s functionality, as well as estimate the typical time-scales of the exploration of the state-space of B-cell traits. Beyond the biological motivations of antibody affinity maturation modeling, the analysis of this learning process leads us to build a mathematical model which could be relevant to model other evolutionary systems, but also gossip or virus propagation. Our method is based on the complementarity between probabilistic tools and numerical simulations.