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.