In this chapter we investigate the connectivity structure of preferential attachment models. We start by discussing an important tool: exchangeable random variables and their distribution described in de Finetti’s Theorem. We apply these results to Pólya urn schemes, which, in turn, we use to describe the distribution of the degrees in preferential attachment models. It turns out that Pólya urn schemes can also be used to describe the local limit of preferential attachment models. A crucial ingredient is the fact that the edges in the Pólya urn representation are conditionally independent, given the appropriate randomness. The resulting local limit is the Pólya point tree, a specific multi-type branching process with continuous types.

In this chapter we draw motivation from real-world networks and formulate random graph models for them. We focus on some of the models that have received the most attention in the literature, namely, Erdos–Rényi random graphs, inhomogeneous random graphs, configuration models, and preferential attachment models. We follow Volume 1, both for the motivation as well as for the introduction of the random graph models involved. Furthermore, we add some convenient additional results, such as degree-truncation for configuration models and switching techniques for uniform random graphs with prescribed degrees. We also discuss preliminaries used in the book, for example concerning power-law distributions.

In this chapter we investigate the local limit of the configuration model, we identify when it has a giant component and find its size and degree structure. We give two proofs, one based on a “the giant is almost local” argument, and another based on a continuous-time exploration of the connected components in the configuration model. Further results include its connectivity transition.

This chapter consists of basic real algebraic geometry. In Chapter 2 we studied basic properties of curves in the plane. We now generalize these properties to arbitrary varieties in any real space and introduce additional properties.

We begin the chapter by briefly discussing polynomial ideals. We then study the dimension of a real variety, connected components, irreducible components, tangent spaces, singular points, regular points, and other properties. We discuss why there is no one well-behaved definition for the degree of a real variety. We also study the projection of real varieties and partitioning polynomials in higher dimensions.

To describe coauthorship networks, we begin with the Erdös number, which links mathematicians to their famously prolific colleague through the papers they have collaborated on. Coauthorship networks help us capture collaborative patterns and identify important features that characterize them. We can also use them to predict how many collaborators a scientist will have in the future based on her coauthorship history. We find that collaboration networks are scale-free, following a power-law distribution. As a consequence of the Matthew effect, frequent collaborators are more likely to collaborate, becoming hubs in their networks. We then explore the small-world phenomenon evidenced in coauthorship networks, which is sometimes referred to as “six degrees of separation.” To understand how a network’s small-worldliness impacts creativity and success, we look to teams of artists collaborating on Broadway musicals, finding that teams perform best when the network they inhabit is neither too big or too small. We end by discussing how connected components within networks provide evidence for the “invisible college.”