We now take a brief detour from the main thread of the book to explore a particular range of examples of the theory we've developed so far, culminating in a discussion of the PageRank algorithm. This gives us to an opportunity to learn the basics of graph theory and probability theory, before covering discrete dynamical systems and stochastic matrices, leading to the PageRank algorithm.

In Section 7.1 we introduce the basic concepts of graph theory, and we show how to represent graphs with matrices. In Section 7.2 we introduce the basic concepts of probability theory, including some facts about expectation values of random variables. In Section 7.3 we present a common kind of application of eigenvalues and eigenvectors to real-world problems in the form of discrete dynamical systems. We refine this application to specific problems involving probabilities in Section 7.4, including searching the Internet using Google’s PageRank algorithm.