In recent years, deep learning has made a significant impact in various fields – helping to push the state-of-the-art forward in many application domains. Convolutional Neural Networks (CNN) have been applied successfully to tasks such as visual object detection, image super-resolution, and video action recognition while Long Short-term Memory (LSTM) and Transformer networks have been used to solve a variety of challenging tasks in natural language processing. However, these popular deep learning architectures (i.e., CNNs, LSTMs, and Transformers) can only handle data that can be represented as grids or sequences. Due to this limitation, many existing deep learning approaches do not generalize to problem domains where the data is represented as graphs – social networks in social network analysis or molecular graphs in chemoinformatics, for instance. The goal of this thesis is to help bridge the gap by studying deep learning solutions that can handle graph data naturally. In particular, we explore deep learning-based approaches in the following areas. 1. Graph Attention. In the real-world, graphs can be both large – with many complex patterns – and noisy which can pose a problem for effective graph mining. An effective way to deal with this issue is to use an attention-based deep learning model. An attention mechanism allows the model to focus on task-relevant parts of the graph which helps the model make better decisions. We introduce a model for graph classification which uses an attention-guided walk to bias exploration towards more task-relevant parts of the graph. For the task of node classification, we study a different model – one with an attention mechanism which allows each node to select the most task-relevant neighborhood to integrate information from. 2. Graph Representation Learning. Graph representation learning seeks to learn a mapping that embeds nodes, and even entire graphs, as points in a low-dimensional continuous space. The function is optimized such that the geometric distance between objects in the embedding space reflect some sort of similarity based on the structure of the original graph(s). We study the problem of learning time-respecting embeddings for nodes in a dynamic network. 3. Brain Network Discovery. One of the fundamental tasks in functional brain analysis is the task of brain network discovery. The brain is a complex structure which is made up of various brain regions, many of which interact with each other. The objective of brain network discovery is two-fold. First, we wish to partition voxels – from a functional Magnetic Resonance Imaging scan – into functionally and spatially cohesive regions (i.e., nodes). Second, we want to identify the relationships (i.e., edges) between the discovered regions. We introduce a deep learning model which learns to construct a group-cohesive partition of voxels from the scans of multiple individuals in the same group. We then introduce a second model which can recover a hierarchical set of brain regions, allowing us to examine the functional organization of the brain at different levels of granularity. Finally, we propose a model for the problem of unified and group-contrasting edge discovery which aims to discover discriminative brain networks that can help us to better distinguish between samples from different classes.

In recent years, deep learning has made a significant impact in various fields – helping to push the state-of-the-art forward in many application domains. Convolutional Neural Networks (CNN) have been applied successfully to tasks such as visual object detection, image super-resolution, and video action recognition while Long Short-term Memory (LSTM) and Transformer networks have been used to solve a variety of challenging tasks in natural language processing. However, these popular deep learning architectures (i.e., CNNs, LSTMs, and Transformers) can only handle data that can be represented as grids or sequences. Due to this limitation, many existing deep learning approaches do not generalize to problem domains where the data is represented as graphs – social networks in social network analysis or molecular graphs in chemoinformatics, for instance. The goal of this thesis is to help bridge the gap by studying deep learning solutions that can handle graph data naturally. In particular, we explore deep learning-based approaches in the following areas. 1. Graph Attention. In the real-world, graphs can be both large – with many complex patterns – and noisy which can pose a problem for effective graph mining. An effective way to deal with this issue is to use an attention-based deep learning model. An attention mechanism allows the model to focus on task-relevant parts of the graph which helps the model make better decisions. We introduce a model for graph classification which uses an attention-guided walk to bias exploration towards more task-relevant parts of the graph. For the task of node classification, we study a different model – one with an attention mechanism which allows each node to select the most task-relevant neighborhood to integrate information from. 2. Graph Representation Learning. Graph representation learning seeks to learn a mapping that embeds nodes, and even entire graphs, as points in a low-dimensional continuous space. The function is optimized such that the geometric distance between objects in the embedding space reflect some sort of similarity based on the structure of the original graph(s). We study the problem of learning time-respecting embeddings for nodes in a dynamic network. 3. Brain Network Discovery. One of the fundamental tasks in functional brain analysis is the task of brain network discovery. The brain is a complex structure which is made up of various brain regions, many of which interact with each other. The objective of brain network discovery is two-fold. First, we wish to partition voxels – from a functional Magnetic Resonance Imaging scan – into functionally and spatially cohesive regions (i.e., nodes). Second, we want to identify the relationships (i.e., edges) between the discovered regions. We introduce a deep learning model which learns to construct a group-cohesive partition of voxels from the scans of multiple individuals in the same group. We then introduce a second model which can recover a hierarchical set of brain regions, allowing us to examine the functional organization of the brain at different levels of granularity. Finally, we propose a model for the problem of unified and group-contrasting edge discovery which aims to discover discriminative brain networks that can help us to better distinguish between samples from different classes.

We present a rigorous yet accessible introduction to structures on finite sets foundational for a formal study of complex networks. This includes a thorough treatment of binary relations, distance spaces, their properties and similarities. Correspondences between relations and graphs are given and a brief introduction to graph theory is followed by a more detailed study of cohesiveness and centrality. We show how graph degeneracy is equivalent to the concept of k-cores, which give a measure of the cohesiveness or interconnectedness of a subgraph. We then further extend this to d-cores of directed graphs. After a brief introduction to topology, focusing on topological spaces from distances, we present a historical discussion on the early developments of algebraic topology. This is followed by a more formal introduction to simplicial homology where we define the homology groups. In the context of algebraic topology, the d-cores of a digraph give rise to a partially ordered set of subgraphs, leading to a set of filtrations that is two-dimensional in nature. Directed clique complexes of digraphs are defined in order to encode the directionality of complete subdigraphs. Finally, we apply these methods to the neuronal network of C.elegans. Persistent homology with respect to directed core filtrations as well as robustness of homology to targeted edge percolations in different directed cores is analyzed. Much importance is placed on intuition and on unifying methods of such dispersed disciplines as sociology and network neuroscience, by rooting them in pure mathematics. / Vi presenterar en rigorös men lättillgänglig introduktion till de abstrakta strukturer på ändliga mängder som är grundläggande för en formell studie av komplexa nätverk. Detta inkluderar en grundlig redogörelse av binära relationer och distansrum, deras egenskaper samt likheter. Korrespondenser mellan olika typer av relationer och grafer förklaras och en kort introduktion till grafteori följs av en mer detaljerad studie av sammanhållning och centralitet. Vi visar hur begreppet 'degeneracy' är ekvivalent med begreppet k-kärnor (eng: k-cores), vilket ger ett mått på sammanhållningen hos en delgraf. Vi utökar sedan detta till konceptet d-kärnor (eng: d-cores) för riktade grafer. Efter en kort introduktion till topologi med fokus på topologiska rum från distansrum, så presenterar vi en historisk diskussion kring den tidiga utvecklingen av algebraisk topologi. Detta följs av en mer formell introduktion till homologi, där vi bl.a. definierar homologigrupperna. Vi definierar sedan så kallade riktade klick-komplex som simplistiska komplex (eng: simplicial complexes) från riktade grafer, där d-kärnorna av en riktad graf då ger upphov till filtrerade komplex i två parametrar. Persistent homologi med avseende på dessa riktade kärnfiltreringar såväl som robusthet mot kantpercolationer i olika kärnor analyseras sedan för det neurala nätverket hos C.Elegans. Stor vikt läggs vid intuition och förståelse, samt vid att förena metodiker för så spridda discipliner som sociologi och neurovetenskap.