This chapter establishes the foundation for network machine learning. We begin with network fundamentals: adjacency matrices, edge directionality, node loops, and edge weights. We then explore node-specific properties such as degree and path length, followed by network-wide metrics including density, clustering coefficients, and average path lengths. The chapter progresses to advanced matrix representations, notably degree matrices and various Laplacian forms, which are crucial for spectral analysis methods. We examine subnetworks and connected components, tools for focusing on relevant network structures. The latter half of the chapter delves into preprocessing techniques. We cover node pruning methods to manage outliers and low-degree nodes. Edge regularization techniques, including thresholding and sparsification, address issues in weighted and dense networks. Finally, we explore edge-weight rescaling methods such as z-score standardization and ranking-based approaches. Throughout, we emphasize practical applications, illustrating concepts with examples and code snippets. These preprocessing steps are vital for addressing noise, sparsity, and computational challenges in network data. By mastering these concepts and techniques, readers will be well-equipped to prepare network data for sophisticated machine learning tasks, setting the stage for the advanced methods presented in subsequent chapters.

This chapter explores deep learning methods for network analysis, focusing on graph neural networks (GNNs) and diffusion-based approaches. We introduce GNNs through a drug discovery case study, demonstrating how molecular structures can be analyzed as networks. The chapter covers GNN architecture, training processes, and their ability to learn complex network representations without explicit feature engineering. We then examine diffusion-based methods, which use random walks to develop network embeddings. These techniques are compared and contrasted with earlier spectral approaches, highlighting their capacity to capture nonlinear relationships and local network structures. Practical implementations using frameworks such as PyTorch Geometric illustrate the application of these methods to large-scale network datasets, showcasing their power in addressing complex network problems across various domains.

This chapter explores advanced applications of network machine learning for multiple networks. We introduce anomaly detection in time series of networks, identifying significant structural changes over time. The chapter then focuses on signal subnetwork estimation for network classification tasks. We present both incoherent and coherent approaches, with incoherent methods identifying edges that best differentiate between network classes, and coherent methods leveraging additional network structure to improve classification accuracy. Practical applications, such as classifying brain networks, are emphasized throughout. These techniques apply to collections of networks, providing a toolkit for analyzing and classifying complex, multinetwork datasets. By integrating previous concepts with new methodologies, we offer a framework for extracting insights and making predictions from diverse network structures with associated attributes.

The theory of kernels offers a rich mathematical framework for the archetypical tasks of classification and regression. Its core insight consists of the representer theorem that asserts that an unknown target function underlying a dataset can be represented by a finite sum of evaluations of a singular function, the so-called kernel function. Together with the infamous kernel trick that provides a practical way of incorporating such a kernel function into a machine learning method, a plethora of algorithms can be made more versatile. This chapter first introduces the mathematical foundations required for understanding the distinguished role of the kernel function and its consequence in terms of the representer theorem. Afterwards, we show how selected popular algorithms, including Gaussian processes, can be promoted to their kernel variant. In addition, several ideas on how to construct suitable kernel functions are provided, before demonstrating the power of kernel methods in the context of quantum (chemistry) problems.

In this chapter, we change our viewpoint and focus on how physics can influence machine learning research. In the first part, we review how tools of statistical physics can help to understand key concepts in machine learning such as capacity, generalization, and the dynamics of the learning process. In the second part, we explore yet another direction and try to understand how quantum mechanics and quantum technologies could be used to solve data-driven task. We provide an overview of the field going from quantum machine learning algorithms that can be run on ideal quantum computers to kernel-based and variational approaches that can be run on current noisy intermediate-scale quantum devices.