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.

A graph $G=(V,E)$ is called an expander if every vertex subset U of size up to $|V|/2$ has an external neighborhood whose size is comparable to $|U|$. Expanders have been a subject of intensive research for more than three decades and have become one of the central notions of modern graph theory. We first discuss the above definition of an expander and its alternatives. Then we present examples of families of expanding graphs and state basic properties of expanders. Next, we introduce a way to argue that a given graph contains a large expanding subgraph. Finally weresearch properties of expanding graphs, such as existence of small separators, of cycles (including cycle lengths),and embedding of large minors.