The chapter gives a brief outline of a variety of random processes associated with networks. It begins with a short introduction to properties of finite Markov chains, including an ergodic theorem and the notion of reversibility, that are useful for understanding networks. Next, the simple vertex random walk is discussed, together with the random walk with restart. The non-backtracking random walk is also introduced, as are some random walks on sets of networks, such as the switching chain and Glauber dynamics. The chapter concludes with a discussion of epidemic models, and of some dynamical systems, such as opinion formation models, on networks.

Data sets that illustrate the methods developed in the book are introduced in this chapter. These are the Florentine marriage networks, Zachary’s karate club network, protein–protein interaction data, collaboration networks, gene co-expression networks, Sampson’s monastery networks, Krackhardt’s manager friendship networks, trade networks, Adamic and Glance’s political blog network and the World Wide Web. Networks encountered in practice are usually derived from larger, more comprehensive collections of data and ways of constructing a network from such data are also discussed.

Stein’s method for distributional approximation was originally developed in the context of normal approximation. It is appropriate for use with network statistics, such as subgraph counts, when the typical counts are large; (compound) Poisson approximation is most useful in sparse networks, normal approximation when networks are denser. In this chapter, Stein’s method for estimating the error in normal approximation, measured with respect to the bounded Wasserstein distance, is outlined. A Stein equation, a first-order ordinary differential equation is derived, and some properties of its solutions are established. Direct application of these results leads to a general bound on the approximation error, which is then applied in the classical context of sums of independent random variables. Then, both local and size bias coupling approaches are developed. The local approach, which uses a double decomposition, is illustrated by application to the number of triangles in the Bernoulli random graph; the calculations are typically more involved than for Poisson approximation. The coupling approach is applied to the distribution of the sample mean in simple random sampling.

The introduction outlines the problems involved in coming to grips with data in the form of a network. A complete pictorial description is possible only for very small networks. More commonly, it is necessary to describe a network in other ways, by choosing appropriate summary statistics, and by comparing it to other, better-understood networks. In many cases, the entire network may be unknown and may change over time, so that only network samples are available for analysis. Problems of interest include conducting statistical analyses when observations are taken from individuals linked by a network, finding communities within a network, estimating how fast an epidemic on a network may be transmitted; and judging to what extent the specification of a network may be in error.

The random geometric graphs considered in this chapter are derived from a configuration of points that are independently placed in an underlying Euclidean space, according to some distribution. Each pair of points that are separated by a distance less than some given threshold is joined by an edge, and the graph then consists only of vertices, corresponding to the points, and of the edges between them, with the positional information discarded. In this model, the edges are no longer independent, and the neighbourhood structure is quite different from the tree-like neighbourhoods in Chapters 11–14; for instance, the average local clustering coefficient is not typically close to zero, even in sparse graphs. A giant component is shown to be unlikely to exist if the density of points is low enough, and to be almost certain to exist if the density of points is high enough, with the ratio of the critical densities fixed as the number of points grows. A subgraph threshold theorem is established, complemented by a number of distributional approximations to the counts of subgraphs; the independence of the positions of the underlying points simplifies this discussion. Under suitable asymptotics, typical shortest path lengths are shown to grow like a power of the number of points, rather than logarithmically, as was the case for the models in Chapters 11–14.

It is possible to define limiting objects that can approximate dense networks, as the number of vertices tend to infinity, in much the same way that the normal distribution approximates that of a sum of independent random variables. In this way, properties of dense graphs can be approximated by those of a suitably chosen limit. A network on n vertices is associated, via its adjacency matrix, with a symmetric function on the unit square that takes values of either 0 or 1, constant over squares of side 1/n, unique up to permutation of the vertex labels. This representation of a network belongs to the larger space of equivalence classes of symmetric, measurable functions on the unit square taking values in the unit interval, with two functions equivalent if one can be obtained from the other by a measure-preserving transformation of the axes; the equivalence classes are called graphons. A metric is defined on the space of graphons with the property that a sequence of ever larger dense networks converges, with respect to this metric, to a limiting graphon W, if, for each k, the distribution, over the set of possible subgraphs of size k, of a randomly chosen induced subgraph of size k in the network converges to a corresponding distribution derived from W.

For the distribution of a count to be close to a Poisson distribution, the events that are counted should typically be almost independent of one another. When counting the number of copies of a small subgraph in a network, it is often the case that the presence of one copy makes it much more likely that other copies share some of the same edges. If this is so, there is a tendency for the copies to occur in clumps. It may well be that the number of clumps has an approximately Poisson distribution, but that the number of copies has a distribution more like that of a sum of a Poisson number of independent random variables, a compound Poisson distribution. Approximation by compound Poisson distributions can also be quantified using Stein’s method. In this chapter, a Stein equation is derived, together with a general estimate of error that is based on it. A local approach is then developed and illustrated using the number of `triangle and whisker’ graphs in the Bernoulli random graph. Another natural question is to address the joint distribution of the counts of a number of different subgraphs in a network. This can be tackled using Stein’s method for Poisson process approximation. The local and coupling approaches to Poisson process approximation are presented and are used in the multivariate context to approximate the joint distribution of short cycles in the Bernoulli random graph.

The structure to be expected in the immediate neighbourhood of a given vertex, in many of the most widely used models of networks, can be matched to that of a suitably chosen branching process. An example of the geometric growth typical of a branching process is furnished by the early stages of the spread of an epidemic disease, as evidenced in the COVID-19 pandemic. The Bernoulli random graph, in turn, can be interpreted as being formally equivalent to the well-established Reed–Frost epidemic model, and the growth of an epidemic more generally can be related to the underlying network of contacts between individuals. In this chapter, branching processes in discrete time are introduced, and their basic properties established. For the Bienaymé–Galton–Watson’ process, moment formulae, the criticality theorem, the distribution of the total population size (in the sub-critical and critical cases) and the asymptotically geometric growth (in the super-critical case) are addressed. In the multitype analogue of the Bienaymé–Galton–Watson’ process, the convergence of the type distribution in the super-critical case is also presented.

The Chung–Lu network model is a multiplicative vertex weighted model, analogous to the Erdős–Rényi mixture model, in which the weights of the vertices may take arbitrary positive values. If the weights are restricted to being drawn from a fixed, finite set, the model is precisely equivalent to an Erdős–Rényi mixture model. Having arbitrary choices of weights allows rather general distributions for the number of neighbours of a vertex. In the Bernoulli random graph, this distribution is constrained to being close to Poisson, and, in particular, cannot have long tails, which makes unlikely that occasional vertices have very large numbers of neighbours, as is often observed in practice. The generality of the possible choices of weights makes the arguments more difficult than for Bernoulli models, since the approximation of the neighbourhoods is no longer as accurate. Despite this, it is possible to match the distribution of local neighbourhoods to those of a suitably chosen branching process, to prove a threshold theorem for the existence of a giant component and a subgraph threshold theorem for the numbers of small subgraphs, and to approximate the distribution of the shortest path length between two vertices having particular weights.

The small world networks considered in this chapter were originally designed to provide examples of networks whose clustering coefficient is not small (as is the case in locally tree-like networks), but in which shortest path lengths grow only logarithmically with the number of vertices if the expected degree of a typical vertex remains bounded (unlike what happens in random geometric graphs). These small world networks can be thought of as interpolating between the Bernoulli random graph and random geometric graphs, by superimposing one upon the other, thus reducing the shortest path lengths without greatly reducing the clustering coefficient. In a limiting version, in which random shortcuts are superimposed on a circle, it is shown that shortest path lengths between randomly chosen points of the circle are divided by a quantity roughly equal to the reciprocal of the number of shortcuts.