The class of hyperbolic groups was introduced and studied in Gromov's paper [125]. In that paper a not necessarily finitely generated group is called hyperbolic if it is hyperbolic as a metric space, in the sense that the space has a hyperbolic inner product, which we define in 6.2.1 below. The group is called word-hyperbolic if it is finitely generated and hyperbolic with respect to the word metric. Since we are only concerned with finitely generated groups in this book, we shall from now on refer to such groups simply as hyperbolic groups.

The fact that there is a large variety of apparently different conditions on a finitely generated group that turn out to be equivalent to hyperbolicity (we present a list of several such conditions in Section 6.6) is itself a strong indication of the fundamental position that these groups occupy in geometric group theory. We have already encountered two of these conditions: groups having a Dehn presentation (or algorithm) in Section 3.5, and strongly geodesically automatic groups in Section 5.8.

Gromov's paper is generally agreed to be difficult to read, but there are several accessible accounts of the basic properties of hyperbolic groups, including those by Alonso et al. [5], Ghys and de la Harpe [94], Bridson and Haefliger [39, Part II, Section Γ] and Neumann and Shapiro [203].

Hyperbolicity conditions

We begin by comparing various notions of hyperbolicity. The definitions that we consider here apply to an arbitrary geodesic metric space, as defined in Section 1.6, but we are mainly interested in the case when the space is the Cayley graph of a finitely generated group.

Let be a geodesic metric space. A geodesic triangle xyz in Γ consists of three points x, y, z together with geodesic paths [xy], [yz] and [zx]. We can define hyperbolicity of Γ in terms of ‘thinness’ properties of geodesic triangles.

In this first chapter, we give an introduction to random graphs and complex networks. We discuss examples of real-world networks and their empirical properties, and give a brief introduction to the kinds of models that we investigate in the book. Further, we introduce the key elements of the notation used throughout the book.

Motivation

The advent of the computer age has incited an increasing interest in the fundamental properties of real-world networks. Due to the increased computational power, large data sets can now easily be stored and investigated, and this has had a profound impact on the empirical studies of large networks. As we explain in detail in this chapter, many real-world networks are small worlds and have large fluctuations in their degrees. These realizations have had fundamental implications for scientific research on networks. Network research is aimed to both to understand why many networks share these fascinating features, and also to investigate what the properties of these networks are in terms of the spread of diseases, routing information and ranking of the vertices present.

The study of complex networks plays an increasingly important role in science. Examples of such networks are electrical power grids and telecommunication networks, social relations, the World-Wide Web and Internet, collaboration and citation networks of scientists, etc. The structure of such networks affects their performance. For instance, the topology of social networks affects the spread of information or disease (see, e.g., Strogatz (2001)). The rapid evolution and success of the Internet have spurred fundamental research on the topology of networks. See Barabási (2002) and Watts (2003) for expository accounts of the discovery of network properties by Barabási, Watts and co-authors. In Newman et al. (2006), you can find some of the original papers detailing the empirical findings of real-world networks and the network models invented for them. The introductory book by Newman (2010) lists many of the empirical properties of, and scientific methods for, networks.

One common feature of complex networks is that they are large. As a result, a global description is utterly impossible, and researchers, both in the applications and in mathematics, have turned to their local description: how many vertices they have, by which local rules vertices connect to one another, etc. These local rules are probabilistic, reflecting the fact that there is a large amount of variability in how connections can be formed