Network analysis in the social sciences developed from a conjuncture of anthropologist's observations about relations in face-to-face groups and mathematical graph theory. A very large part of social network methodology, consequently, deals with relatively small networks, networks where we have confidence in the reliability of our observations about the relations among the actors. Most of the tools of social network analysis involve the use of mathematical functions to describe networks and their sub-structures.
In more recent work, however, some of the focus of social network research has moved away from these roots. Increasingly, the social networks that are being studied may contain many nodes; and, sometimes our observations about these very large networks are based not on censuses, but on samples of nodes. Network researchers have come to recognize that the relations that they study may be constantly evolving, and that the relations observed at one point in time may not the entirely typical because the pattern of relations is not "in equilibrium." They have also recognized that sometimes our observations are fallible -- we fail to record a relation that actually exists, or mis-measure the strength of a tie.
All of these concerns (large networks, sampling, concern about the reliability of observations) have led social network researchers to begin to apply the techniques of descriptive and inferential statistics in their work. Statistics provide useful tools for summarizing large amounts of information, and for treating observations as stochastic, rather than deterministic outcomes of social processes.
Descriptive statistics have proven to be of great value because they provide convenient tools to summarize key facts about the distributions of actors, attributes, and relations; statistical tools can describe not only the shape of one distribution, but also joint distributions, or "statistical association." So, statistical tools have been particularly helpful in describing, predicting, and testing hypotheses about the relations between network properties.
Inferential statistics have also proven to have very useful applications to social network analysis. At a most general level, the question of "inference" is: how much confidence can I have that the pattern I see in the data I've collected is actually typical of some larger population, or that the apparent pattern is not really just a random occurrence?
In this chapter we will look at some of the ways in which quite basic statistical tools have been applied in social network analysis. These are only the starting point. The development of more powerful statistical tools especially tuned for the needs of social network analysis is one of the most rapidly developing "cutting edges" of the field.