The classical wavelet transform has been widely used in image and signal processing, where a signal is decomposed into a combination of basis signals. By analyzing the individual contribution of the basis signals, one can infer properties of the original signal. This dissertation presents an overview of the extension of the classical signal processing theory to graph domains. Specifically, we review the graph Fourier transform and graph wavelet transforms both of which based on the spectral graph theory, and explore their properties through illustrative examples. The main features of the spectral graph wavelet transforms are presented using synthetic and real-world data. Furthermore, we introduce in this dissertation a novel method for visual analysis of dynamic networks, which relies on the graph wavelet theory. Dynamic networks naturally appear in a multitude of applications from different domains. Analyzing and exploring dynamic networks in order to understand and detect patterns and phenomena is challenging, fostering the development of new methodologies, particularly in the field of visual analytics. Our method enables the automatic analysis of a signal defined on the nodes of a network, making viable the detection of network properties. Specifically, we use a fast approximation of the graph wavelet transform to derive a set of wavelet coefficients, which are then used to identify activity patterns on large networks, including their temporal recurrence. The wavelet coefficients naturally encode spatial and temporal variations of the signal, leading to an efficient and meaningful representation. This method allows for the exploration of the structural evolution of the network and their patterns over time. The effectiveness of our approach is demonstrated using different scenarios and comparisons involving real dynamic networks. / A transformada wavelet clássica tem sido amplamente usada no processamento de imagens e sinais, onde um sinal é decomposto em uma combinação de sinais de base. Analisando a contribuição individual dos sinais de base, pode-se inferir propriedades do sinal original. Esta tese apresenta uma visão geral da extensão da teoria clássica de processamento de sinais para grafos. Especificamente, revisamos a transformada de Fourier em grafo e as transformadas wavelet em grafo ambas fundamentadas na teoria espectral de grafos, e exploramos suas propriedades através de exemplos ilustrativos. As principais características das transformadas wavelet espectrais em grafo são apresentadas usando dados sintéticos e reais. Além disso, introduzimos nesta tese um método inovador para análise visual de redes dinâmicas, que utiliza a teoria de wavelets em grafo. Redes dinâmicas aparecem naturalmente em uma infinidade de aplicações de diferentes domínios. Analisar e explorar redes dinâmicas a fim de entender e detectar padrões e fenômenos é desafiador, fomentando o desenvolvimento de novas metodologias, particularmente no campo de análise visual. Nosso método permite a análise automática de um sinal definido nos vértices de uma rede, tornando possível a detecção de propriedades da rede. Especificamente, usamos uma aproximação da transformada wavelet em grafo para obter um conjunto de coeficientes wavelet, que são então usados para identificar padrões de atividade em redes de grande porte, incluindo a sua recorrência temporal. Os coeficientes wavelet naturalmente codificam variações espaciais e temporais do sinal, criando uma representação eficiente e com significado expressivo. Esse método permite explorar a evolução estrutural da rede e seus padrões ao longo do tempo. A eficácia da nossa abordagem é demonstrada usando diferentes cenários e comparações envolvendo redes dinâmicas reais.

Cette thèse propose de nouveaux outils adaptés à l'analyse des réseaux : sociaux, de transport, de neurones, de protéines, de télécommunications... Ces réseaux, avec l'essor de certaines technologies électroniques, informatiques et mobiles, sont de plus en plus mesurables et mesurés ; la demande d'outils d'analyse assez génériques pour s'appliquer à ces réseaux de natures différentes, assez puissants pour gérer leur grande taille et assez pertinents pour en extraire l'information utile, augmente en conséquence. Pour répondre à cette demande, une grande communauté de chercheurs de différents horizons scientifiques concentre ses efforts sur l'analyse des graphes, des outils mathématiques modélisant la structure relationnelle des objets d'un réseau. Parmi les directions de recherche envisagées, le traitement du signal sur graphe apporte un éclairage prometteur sur la question : le signal n'est plus défini comme en traitement du signal classique sur une topologie régulière à n dimensions, mais sur une topologie particulière définie par le graphe. Appliquer ces idées nouvelles aux problématiques concrètes d'analyse d'un réseau, c'est ouvrir la voie à une analyse solidement fondée sur la théorie du signal. C'est précisément autour de cette frontière entre traitement du signal et science des réseaux que s'articule cette thèse, comme l'illustrent ses deux principales contributions. D'abord, une version multiéchelle de détection de communautés dans un réseau est introduite, basée sur la définition récente des ondelettes sur graphe. Puis, inspirée du concept classique de bootstrap, une méthode de rééchantillonnage de graphes est proposée à des fins d'estimation statistique. / This thesis describes new tools specifically designed for the analysis of networks such as social, transportation, neuronal, protein, communication networks... These networks, along with the rapid expansion of electronic, IT and mobile technologies are increasingly monitored and measured. Adapted tools of analysis are therefore very much in demand, which need to be universal, powerful, and precise enough to be able to extract useful information from very different possibly large networks. To this end, a large community of researchers from various disciplines have concentrated their efforts on the analysis of graphs, well define mathematical tools modeling the interconnected structure of networks. Among all the considered directions of research, graph signal processing brings a new and promising vision : a signal is no longer defined on a regular n-dimensional topology, but on a particular topology defined by the graph. To apply these new ideas on the practical problems of network analysis paves the way to an analysis firmly rooted in signal processing theory. It is precisely this frontier between signal processing and network science that we explore throughout this thesis, as shown by two of its major contributions. Firstly, a multiscale version of community detection in networks is proposed, based on the recent definition of graph wavelets. Then, a network-adapted bootstrap method is introduced, that enables statistical estimation based on carefully designed graph resampling schemes.

Tout d'abord, nous présentons des méthodes permettant d'inférer un graphe à partir de signaux, afin de modéliser le support des données à classifier. Ensuite, des translations préservant les voisinages des sommets sont identifiées sur le graphe inféré. Enfin, ces translations sont utilisées pour déplacer un noyau convolutif sur le graphe, afin dedéfinir un réseau de neurones convolutif adapté aux données d'entrée.Nous avons illustré notre méthodologie sur une base de données d'images. Sans utiliser de connaissances sur les signaux, nous avons pu inférer un graphe proche d'une grille. Les translations sur ce graphe sont proches des translations Euclidiennes, ce qui nous a permis de définir un réseau de neurones convolutif très similaire à ce que l'on aurait pu obtenir en utilisant l'information que les signaux sont des images. Ce réseau, entraîné sur les données initiales, a dépassé lesperformances des méthodes de l'état de l'art de plus de 13 points, tout en étant simple et facilement améliorable.La méthode que nous avons introduite est une généralisation des réseaux de neurones convolutifs, car ceux-ci sont des cas particuliers de notre approche quand le graphe est une grille. Nos travaux ouvrent donc de nombreuses perspectives, car ils fournissent une méthode efficace pour construire des réseaux adaptés aux données. / This manuscript sums up our work on extending convolutional neuralnetworks to irregular domains through graph inference. It consists of three main chapters, each giving the details of a part of a methodology allowing the definition of such networks to process signals evolving on graphs with unknown structures.First, graph inference from data is explored, in order to provide a graph modeling the support of the signals to classify. Second, translation operators that preserve neighborhood properties of the vertices are identified on the inferred graph. Third, these translations are used to shift a convolutional kernel on the graph in order to define a convolutional neural network that is adapted to the input data.We have illustrated our methodology on a dataset of images. While not using any particular knowledge on the signals, we have been able to infer a graph that is close to a grid. Translations on this graph resemble Euclidean translations. Therefore, this has allowed us to define an adapted convolutional neural network that is very close what one would obtain when using the information that signals are images. This network, trained on the initial data, has out performed state of the art methods by more than 13 points, while using a very simple and easily improvable architecture.The method we have introduced is a generalization of convolutional neural networks. As a matter of fact, they can be seen as aparticularization of our approach in the case where the graph is a grid. Our work thus opens the way to numerous perspectives, as it provides an efficient way to build networks that are adapted to the data.

Signal processing is used in a wide variety of applications, ranging from digital image processing to biomedicine. Recently, some tools from signal processing have been extended to the context of graphs, allowing its use on irregular domains. Among others, the Fourier Transform and the Wavelet Transform have been adapted to such context. Graph signal processing (GSP) is a new field with many potential applications on data exploration. In this dissertation we show how tools from graph signal processing can be used for visual analysis. Specifically, we proposed a data filtering method, based on spectral graph filtering, that led to high quality visualizations which were attested qualitatively and quantitatively. On the other hand, we relied on the graph wavelet transform to enable the visual analysis of massive time-varying data revealing interesting phenomena and events. The proposed applications of GSP to visually analyze data are a first step towards incorporating the use of this theory into information visualization methods. Many possibilities from GSP can be explored by improving the understanding of static and time-varying phenomena that are yet to be uncovered. / O processamento de sinais é usado em uma ampla variedade de aplicações, desde o processamento digital de imagens até a biomedicina. Recentemente, algumas ferramentas do processamento de sinais foram estendidas ao contexto de grafos, permitindo seu uso em domínios irregulares. Entre outros, a Transformada de Fourier e a Transformada Wavelet foram adaptadas nesse contexto. O Processamento de Sinais em Grafos (PSG) é um novo campo com muitos aplicativos potenciais na exploração de dados. Nesta dissertação mostramos como ferramentas de processamento de sinal gráfico podem ser usadas para análise visual. Especificamente, o método de filtragem de dados porposto, baseado na filtragem de grafos espectrais, levou a visualizações de alta qualidade que foram atestadas qualitativa e quantitativamente. Por outro lado, usamos a transformada de wavelet em grafos para permitir a análise visual de dados massivos variantes no tempo, revelando fenômenos e eventos interessantes. As aplicações propostas do PSG para analisar visualmente os dados são um primeiro passo para incorporar o uso desta teoria nos métodos de visualização da informação. Muitas possibilidades do PSG podem ser exploradas melhorando a compreensão de fenômenos estáticos e variantes no tempo que ainda não foram descobertos.

Signal processing is used in a wide variety of applications, ranging from digital image processing to biomedicine. Recently, some tools from signal processing have been extended to the context of graphs, allowing its use on irregular domains. Among others, the Fourier Transform and the Wavelet Transform have been adapted to such context. Graph signal processing (GSP) is a new field with many potential applications on data exploration. In this dissertation we show how tools from graph signal processing can be used for visual analysis. Specifically, we proposed a data filtering method, based on spectral graph filtering, that led to high quality visualizations which were attested qualitatively and quantitatively. On the other hand, we relied on the graph wavelet transform to enable the visual analysis of massive time-varying data revealing interesting phenomena and events. The proposed applications of GSP to visually analyze data are a first step towards incorporating the use of this theory into information visualization methods. Many possibilities from GSP can be explored by improving the understanding of static and time-varying phenomena that are yet to be uncovered. / O processamento de sinais é usado em uma ampla variedade de aplicações, desde o processamento digital de imagens até a biomedicina. Recentemente, algumas ferramentas do processamento de sinais foram estendidas ao contexto de grafos, permitindo seu uso em domínios irregulares. Entre outros, a Transformada de Fourier e a Transformada Wavelet foram adaptadas nesse contexto. O Processamento de Sinais em Grafos (PSG) é um novo campo com muitos aplicativos potenciais na exploração de dados. Nesta dissertação mostramos como ferramentas de processamento de sinal gráfico podem ser usadas para análise visual. Especificamente, o método de filtragem de dados porposto, baseado na filtragem de grafos espectrais, levou a visualizações de alta qualidade que foram atestadas qualitativa e quantitativamente. Por outro lado, usamos a transformada de wavelet em grafos para permitir a análise visual de dados massivos variantes no tempo, revelando fenômenos e eventos interessantes. As aplicações propostas do PSG para analisar visualmente os dados são um primeiro passo para incorporar o uso desta teoria nos métodos de visualização da informação. Muitas possibilidades do PSG podem ser exploradas melhorando a compreensão de fenômenos estáticos e variantes no tempo que ainda não foram descobertos.

The core research of this thesis, mostly comprising chapter four, has been accepted to the Learning on Graphs (LoG) 2022 conference for a spotlight presentation as a standalone paper, under the title "Taxonomy of Benchmarks in Graph Representation Learning", and is to be published in the Proceedings of Machine Learning Research (PMLR) series. As a main author of the paper, my specific contributions to this paper cover problem formulation, design and implementation of our taxonomy framework and experimental pipeline, collation of our results and of course the writing of the article. / L'apprentissage profond sur les graphes a atteint des niveaux de succès sans précédent ces dernières années grâce aux réseaux de neurones de graphes (GNN), des architectures de réseaux de neurones spécialisées qui ont sans équivoque surpassé les approches antérieurs d'apprentissage définies sur des graphes. Les GNN étendent le succès des réseaux de neurones aux données structurées en graphes en tenant compte de leur géométrie intrinsèque. Bien que des recherches approfondies aient été effectuées sur le développement de GNN avec des performances supérieures à celles des modèles références d'apprentissage de représentation graphique, les procédures d'analyse comparative actuelles sont insuffisantes pour fournir des évaluations justes et efficaces des modèles GNN. Le problème peut-être le plus répandu et en même temps le moins compris en ce qui concerne l'analyse comparative des graphiques est la "couverture de domaine": malgré le nombre croissant d'ensembles de données graphiques disponibles, la plupart d'entre eux ne fournissent pas d'informations supplémentaires et au contraire renforcent les biais potentiellement nuisibles dans le développement d’un modèle GNN. Ce problème provient d'un manque de compréhension en ce qui concerne les aspects d'un modèle donné qui sont sondés par les ensembles de données de graphes. Par exemple, dans quelle mesure testent-ils la capacité d'un modèle à tirer parti de la structure du graphe par rapport aux fonctionnalités des nœuds? Ici, nous développons une approche fondée sur des principes pour taxonomiser les ensembles de données d'analyse comparative selon un "profil de sensibilité" qui est basé sur la quantité de changement de performance du GNN en raison d'une collection de perturbations graphiques. Notre analyse basée sur les données permet de mieux comprendre quelles caractéristiques des données de référence sont exploitées par les GNN. Par conséquent, notre taxonomie peut aider à la sélection et au développement de repères graphiques adéquats et à une évaluation mieux informée des futures méthodes GNN. Enfin, notre approche et notre implémentation dans le package GTaxoGym (https://github.com/G-Taxonomy-Workgroup/GTaxoGym) sont extensibles à plusieurs types de tâches de prédiction de graphes et à des futurs ensembles de données. / Deep learning on graphs has attained unprecedented levels of success in recent years thanks to Graph Neural Networks (GNNs), specialized neural network architectures that have unequivocally surpassed prior graph learning approaches. GNNs extend the success of neural networks to graph-structured data by accounting for their intrinsic geometry. While extensive research has been done on developing GNNs with superior performance according to a collection of graph representation learning benchmarks, current benchmarking procedures are insufficient to provide fair and effective evaluations of GNN models. Perhaps the most prevalent and at the same time least understood problem with respect to graph benchmarking is "domain coverage": Despite the growing number of available graph datasets, most of them do not provide additional insights and on the contrary reinforce potentially harmful biases in GNN model development. This problem stems from a lack of understanding with respect to what aspects of a given model are probed by graph datasets. For example, to what extent do they test the ability of a model to leverage graph structure vs. node features? Here, we develop a principled approach to taxonomize benchmarking datasets according to a "sensitivity profile" that is based on how much GNN performance changes due to a collection of graph perturbations. Our data-driven analysis provides a deeper understanding of which benchmarking data characteristics are leveraged by GNNs. Consequently, our taxonomy can aid in selection and development of adequate graph benchmarks, and better informed evaluation of future GNN methods. Finally, our approach and implementation in the GTaxoGym package (https://github.com/G-Taxonomy-Workgroup/GTaxoGym) are extendable to multiple graph prediction task types and future datasets.