In this chapter we've taken a look at some of the most basic and common approaches to applying statistical analysis to the attributes of actors embedded in networks, the relations among these actors, and the similarities between multiple relational networks connecting the same actors. We've covered a lot of ground. But, there is still a good bit more, as the application of statistical modeling to network data is one of the "leading edges" of the field of social (and other) network analyses.
There are two main reasons for the interest in applying statistics to what was, originally, deterministic graph theory from mathematics. First, for very large networks, methods for finding and describing the distributions of network features provide important tools for understanding the likely patterns of behavior of the whole network and the actors embedded in it. Second, we have increasingly come to realize that the relations we see among actors in a network at a point in time are best seen as probabilistic ("stochastic") outcomes of underlying processes of evolution of networks, and probabilistic actions of actors embedded in those networks. Statistical methods provide ways of dealing with description and hypothesis testing that take this uncertainty into account.
We've reviewed methods for examining relations between two (or more) graphs involving the same actors. These tools are particularly useful for trying to understand multi-plex relations, and for testing hypotheses about how the pattern of relations in one whole network relate to the pattern of relations in another.
We've also looked at tools that deal individual nodes. These tools allow us to examine hypotheses about the relational and non-relational attributes of actors, and to draw correct inferences about relations between variables when the observations (actors) are not independent.
And, we've taken a look at a variety of approaches that relate attributes of actors to their positions in networks. Much of the focus here is on how attributes may pattern relations (e.g. homophily), or how network closeness of distance may affect similarity of attributes (or vice versa).
Taken together, the marriage of statistics and mathematics in social network analysis has already produced some very useful ways of looking at patterns of social relations. It is likely that this interface will be one of the areas of most rapid development in the field of social network methods in the coming years.