The chapter collects the information obtained in the earlier chapters about network summary statistics under some of the best-studied models. The statistics considered are the degree distribution, the numbers of vertices of given degree, counts of subgraphs, the structure of neighbourhoods, the numbers of vertices of different types in the giant component and shortest path lengths.

In this chapter, the mathematically simplest random graph, the Bernoulli random graph, is introduced. Each of the possible edges is present, independently, with the same probability, so that the model is one of a network entirely without structure. To start with, the structure of the graph in the neighbourhood of a point is investigated and is shown to be very similar to that of a branching process with Poisson-distributed offspring numbers. Explicit bounds on the accuracy of the approximation are derived, using the Poisson approximation techniques derived in Chapter 7. The classical threshold theorem for the existence of a giant component is then established; the precision of the neighbourhood approximation simplifies the proof. The counts of small subgraphs are then investigated, and a subgraph threshold theorem is proved. Finally, the distribution of the length (in graph distance) of the shortest path between two vertices is investigated. These grow logarithmically with the number of points, if the expected degree of a vertex is kept constant. Once again, the approximation of the neighbourhood structure is a key element in the proofs, and the statement of the main theorem involves the Laplace transform of the distribution of the limit random variable associated with the approximate branching process.

Many networks are not completely known; the only access to them is by taking samples. This chapter presents methods for deducing information about the whole network from samples. First, some classical sampling methods are briefly considered; random sampling, with and without replacement, stratified sampling and the Horvitz–Thompson estimator. Then sampling methods based on the network structure are introduced, including two-level sampling, induced subgraph sampling, star and snowball sampling and traversal sampling. The differences between the structure of sample networks and those of the parent network are illustrated for some simple models. Finally, the problem of assessing whether a particular network sample is `interesting’ is discussed; interesting, in that it differs from what might be expected of a typical network sample.

There are many topics that are not covered in the book. First, networks may be weighted, directed or signed. Then networks may exhibit structures other than those considered in the book, such as hierarchical structures, or have edges of different types, and collections of networks may arise as snapshots of a network process evolving in time. Each of these settings requires different methods of analysis. Then relationships may be expressed in more intricate ways; `edges’ may link more than just two objects, as in a hypergraph, and abstract simplicial complexes can be thought of as higher dimensional analogues of geometric graphs. These, and other topics, are sketched in this chapter. The material in this book forms a general basis that can be used in coming to grips with these more advanced settings.

The configuration and GPDS models allow the degrees of the vertices to be exactly specified. They are mathematically more challenging than those considered in Chapters 11–13, because the assignment of edges is no longer independent. The configuration model, which generates multigraphs, acts as a more tractable alternative to the GPDS model, because of the symmetries inherent in the random mapping that is used to define it, and because each realization from the configuration model that is simple is also a realization from the GPDS with the same vertex degrees. In this chapter, the neighbourhood structure in the configuration model is approximated by that of a related branching process, and a threshold theorem for the appearance of a giant component is also derived. For subgraph counts, the configuration multigraphs are first made simple, by collapsing multiple edges and deleting self-loops; the resulting graphs are shown to satisfy a subgraph threshold theorem. Finally, the neighbourhood structure is used to derive an approximation to the distribution of the length of the shortest path between two vertices. The proofs in this chapter are very much more involved than those elsewhere in the book.

The chapter introduces some of the statistics most commonly used to describe networks. These can be seen as analogues, in a network context, of quantities such as mean and standard deviation for a sample of real numbers. They can be roughly divided into two categories: topological summaries, such as the collection of degrees of the vertices of the network, that could be derived from a picture of the network, and spectral summaries, such as the eigenvector centrality, that are derived from the spectral decomposition of the adjacency matrix of the network (or of one or more related matrices). Many of them, such as clustering coefficients, can be formulated as local summaries, computed at each vertex of the network and can then be combined to yield a global value characteristic of the entire network. The results of a number of the summaries are compared with one another, using the Florentine marriage network as an example.

This chapter discusses a number of generalizations of the Bernoulli random graph. The first is to Erdős–Rényi mixture models, in which each vertex belongs to one of a small number of distinct types, and the probability of an edge between two vertices being present depends on the types of the two vertices. Much as for the Bernoulli random graph, the neighbourhood structure is approximated by a multitype branching process. A threshold theorem for the existence of a giant component is derived, and the asymptotic proportions of the vertices in the giant component that belong to the different types is determined. Finally, the distribution of the length of the shortest path between two vertices of given types is approximated. The properties of some special network models of this kind, vertex weighted random graphs and bipartite networks, are examined. Another set of models considered is that of directed Bernoulli random graphs, in which there may be edges in either (or both) directions between any pair of vertices. Finally, Poissonized Erdős–Rényi mixture models for multigraphs are introduced, in which edges may be present between any pair of vertices, and the number of such edges has a Poisson distribution, with mean depending on the types of the vertices, the numbers being chosen independently. The extent to which these models differ from the standard Erdős–Rényi mixture models in the sparse regime is discussed.

This chapter considers settings in which networks are used in statistical inference. First, there is often dependence between measurements taken at vertices of a network, with vertices at small graph distance more highly dependent. Methods for accommodating such dependence, including network effects and network disturbance models, are discussed. In such cases, the network is treated as a nuisance parameter. In contrast, the network itself may be the object of primary interest. Fitting a model to the network may make possible inference, for instance about the presence or absence of a particular edge, or of characteristics of vertices; this is valuable if knowledge of the network is not perfect. If more than one network is available, it may also be possible to compare the properties of different, but related networks, using the fitted models, and to use the comparisons to uncover scientifically interesting features. It can also be of interest to find ways of comparing whole networks with one another, when they may have very different vertex sets, but are conjectured to be related; methods discussed include the graphlet correlation distance, Netdis and NetEMD. For instance, the similarities between the protein–protein interaction networks of different species can be used as a basis for constructing phylogenies.