This chapter presents a framework for learning useful representations, or embeddings, of networks. Building on the statistical models from Chapter 4, we explore techniques to transform complex network data into vector representations suitable for traditional machine learning algorithms. We begin with maximum likelihood estimation for simple network models, then motivate the need for network embeddings by contrasting network dependencies with typical machine learning independence assumptions. We progress through spectral embedding methods, introducing adjacency spectral embedding (ASE) for learning latent position representations from adjacency matrices, and Laplacian spectral embedding (LSE) as an alternative approach effective for networks with degree heterogeneities. The chapter then extends to multiple network representations, exploring parallel techniques like omnibus embedding (OMNI) and fused methods such as multiple adjacency spectral embedding (MASE). We conclude by addressing the estimation of appropriate latent dimensions for embeddings. Throughout, we emphasize practical applications with code examples and visualizations. This unified framework for network embedding enables the application of various machine learning algorithms to network analysis tasks, bridging complex network structures and traditional data analysis techniques.

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.