We shall begin the chapter by explaining what PageRank is and how it is computed efficiently. Yet the war between those who want to make the Web useful and those who would exploit it for their own purposes is never over. When PageRank was established as an essential technique for a search engine, spammers invented ways to manipulate the PageRank of a Web page, often called link spam. That development led to the response of TrustRank and other techniques for preventing spammers from attacking PageRank. We shall discuss TrustRank and other approaches to detecting link spam. Finally, this chapter also covers some variations on PageRank. These techniques include topic-sensitive PageRank (which can also be adapted for combating link spam) and the HITS, or “hubs and authorities” approach to evaluating pages on the Web.

We include in this chapter a discussion of generalizations of MapReduce, first to systems that support acyclic workflows and then to systems that implement recursive algorithms. Our last topic for this chapter is the design of good MapReduce algorithms, a subject that often differs significantly from the matter of designing good parallel algorithms to be run on a supercomputer. When designing MapReduce algorithms, we often find that the greatest cost is in the communication. We thus investigate communication cost and what it tells us about the most efficient MapReduce algorithms. For several common applications of MapReduce we are able to give families of algorithms that optimally trade the communication cost against the degree of parallelism.

We begin our discussion of locality-sensitive hashing (LSH) with an examination of the problem of finding similar documents – those that share a lot of common text. We first show how to convert documents into sets in a way that lets us view textual similarity of documents as sets having a large overlap. A second key trick we need is minhashing, which is a way to convert large sets into much smaller representations, called signatures, that still enable us to estimate closely the Jaccard similarity of the represented sets. Finally, we see how to apply the bucketing idea inherent in LSH to the signatures. In Section 3.5 we begin our study of how to apply LSH to items other than sets. We consider the general notion of a distance measure that tells to what degree items are similar. Then, we consider the general idea of locality-sensitive hashing, and we see how to do LSH for some data types other than sets. We examine in detail several applications of the LSH idea. Finally, we consider some techniques for finding similar sets that can be more efficient than LSH when the degree of similarity we want is very high.

Clustering is the process of examining a collection of “points,” and grouping the points into “clusters” according to some distance measure. The goal is that points in the same cluster have a small distance from one another, while points in different clusters are at a large distance from one another. A suggestion of what clusters might look like was seen in Fig. 1.1. However, there the intent was that there were three clusters around three different road intersections, but two of the clusters blended into one another because they were not sufficiently separated. Our goal in this chapter is to offer methods for discovering clusters in data. We are particularly interested in situations where the data is very large, and/or where the space either is high-dimensional, or the space is not Euclidean at all. We shall therefore discuss several algorithms that assume the data does not fit in main memory. However, we begin with the basics: the two general approaches to clustering and the methods for dealing with clusters in a non-Euclidean space.

We begin with the essence of data mining and a discussion of how data mining is treated by the various disciplines that contribute to this field. We cover “Bonferroni’s Principle,” which is really a warning about overusing the ability to mine data. We also summarize a few useful ideas that are not data mining per se, but are useful in understanding some important data-mining concepts. These include the TF.IDF measure of word importance, behavior of hash functions and indexes, and identities involving e, the base of natural logarithms. Finally, we give an outline of the topics covered in the balance of the book.

In this chapter we shall explore the idea of dimensionality reduction in more detail. We begin with a discussion of eigenvalues and their use in “principal component analysis” (PCA). We cover singular-value decomposition, a more powerful version of UV-decomposition. Finally, because we are always interested in the largest data sizes we can handle, we look at another form of decomposition, called CUR-decomposition, which is a variant of singular-value decomposition that keeps the matrices of the decomposition sparse if the original matrix is sparse.