Many researchers have shown, using empirical studies, that social network data possess strong deviations from randomness. That is, when one analyzes such data using baseline or null models that assume various types of randomness and specific tendencies that should arise in such data (such as equal popularity, lack of transitivity, or no reciprocity), the data often fail to agree with predictions from the models. Other researchers have reasoned that these deviations from randomness in social network data are caused by the presence of special structural patterns (such as differential popularity, transitivity, or tendencies toward reciprocity of relations) that have been studied for years by social network theorists. In Chapter 6 we described a few of these theories; in this chapter, we show how some of these theories can be tested by studying triads using the triad census (the counts of the various types of triads).

For example, consider transitivity, as defined in Chapter 6. This theory states that various triads are not possible, or at least should not occur, if actor behaviors are transitive. Certain triads should occur if behavior is indeed transitive. Suppose that a researcher has a network under investigation, and wishes to study whether this proposition is viable. We can take the triads that actually arise in the network, and compare these observed frequencies to the frequencies that are to be expected. The details of this comparison will be given in this chapter. For such comparisons, we will need some of the random directed graph distributions described in Chapter 13.