In this chapter, an adaptation of Stein’s method for bounding the error in multivariate normal approximation is presented. For simplicity, the distance measure used is based on expectations of functions with three bounded derivatives; more natural measures of distance would require much more complicated treatments. The Stein equation used is now a second-order partial differential equation. Solutions to the equation are exhibited, and some of their properties are established; they can then be used to derive a general bound on the approximation error in multivariate normal approximation. For exploiting the general bound, a local approach is introduced, which uses a multivariate version of the double decomposition used for (univariate) normal approximation. This is applied to the number of monochrome edges in a graph whose vertices are randomly coloured. A size bias coupling approach is also developed and applied to the joint distribution of counts of vertices of different degrees in the Bernoulli random graph.

In most network models, the distributions of summary statistics are rarely known exactly. However, it may well be possible to approximate their distributions by other, well-known distributions, when the values of local statistics at different vertices are only weakly dependent. Such settings are well suited to the application of the Stein–Chen method, which, in the context of Poisson approximation, enables concrete estimates of the approximation error to be derived, with respect to the total variation distance. In this chapter, the Stein–Chen method is developed in some detail. A Stein equation is derived, together with the necessary properties of its solutions, and a general estimate of approximation error is given, which is expressed solely in terms of the random variable whose distribution is being approximated. The method is applied to sums of dependent random variables, using both a local and a coupling approach. Examples given include the number of triangles and the number of isolated vertices in a Bernoulli random graph.

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.

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.