The fifth chapter explores the application of spectral graph theory to network data analysis. The chapter begins with an introduction to fundamental graph theory concepts, including undirected and directed graphs, graph connectivity, and matrix representations such as the adjacency and Laplacian matrices. It then discusses the variational characterization of eigenvalues and their significance in understanding the structure of graphs. The chapter highlights the spectral properties of the Laplacian matrix, particularly its role in graph connectivity and partitioning. Key applications, such as spectral clustering for community detection and the analysis of random graph models like Erdős–Rényi random graphs and stochastic blockmodels, are presented. The chapter concludes with a detailed exploration of graph partitioning algorithms and their practical implementations using Python.

This is an introduction to representation theory and harmonic analysis on finite groups. This includes, in particular, Gelfand pairs (with applications to diffusion processes à la Diaconis) and induced representations (focusing on the little group method of Mackey and Wigner). We also discuss Laplace operators and spectral theory of finite regular graphs. In the last part, we present the representation theory of GL(2, Fq), the general linear group of invertible 2 × 2 matrices with coefficients in a finite field with q elements. More precisely, we revisit the classical Gelfand–Graev representation of GL(2, Fq) in terms of the so-called multiplicity-free triples and their associated Hecke algebras. The presentation is not fully self-contained: most of the basic and elementary facts are proved in detail, some others are left as exercises, while, for more advanced results with no proof, precise references are provided.

In this chapter, we develop spectral techniques. We highlight some applications to Markov chain mixing and network analysis. The main tools are the spectral theorem and the variational characterization of eigenvalues, which we review together with some related results. We also give a brief introduction to spectral graph theory and detail an application to community recovery. Then we apply the spectral theorem to reversible Markov chains. In particular we define the spectral gap and establish its close relationship to the mixing time. We also show in that the spectral gap can be bounded using certain isoperimetric properties of the underlying network. We prove Cheeger’s inequality, which quantifies this relationship, and introduce expander graphs, an important family of graphs with good “expansion.” Applications to mixing times are also discussed. One specific technique is the “canonical paths method,” which bounds the spectral graph by formalizing a notion of congestion in the network.

In Chapter 6 we saw several applications of polynomial methods in finite fields. In this chapter, we continue our study of finite fields, by studying point-line incidences in a finite plane. Much less is known about incidences over finite fields and many incidence problems become more difficult in this case. Unlike incidences over the reals, there is no one main technique that leads to most of the current best bounds. Instead, each bound that we derive in this chapter requires a rather different set of tools.

For the proofs in this chapter, we introduce tools such as the projective plane, eigenvalues of a graph, and more. We also use finite field incidence bounds to study the finite field sum-product problem.