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 small-world structure in rank-1 and general inhomogeneous random graphs. For this, we develop path-counting techniques that are interesting in their own right.

In this chapter we introduce the general setting of inhomogeneous random graphs that are generalizations of the Erdos–Rényi and generalized random graphs. In inhomogeneous random graphs, the status of edges is independent with unequal edge-occupation probabilities. While these edge probabilities are moderated by vertex weights in generalized random graphs, in the general setting they are described in terms of a kernel. The main results in this chapter concern the degree structure, the multi-type branching process local limits, and the phase transition in these inhomogeneous random graphs. We also discuss various examples, and indicate that they can have rather different structure.