As I have mentioned already, dyads are the elementary units of analysis in the network. Triads are the elementary units of structural analysis in the network. Let's look and see what information we can gather from dyads and triads. Let's start by thinking of dyads and triads in an applied sense. What is the essential difference between dyads and triads? Sociology hubs, Georg Simmel, in late 19th century, wrote extensively about the difference between a dyad at two member group and the triad, which is a three member group, that is a publication from 1902. In the former, if one person withdraws, the group can no longer exist. We can think of a divorce, for example, which effectively ends the group if the married couple decides to separate. In a triad, however, the dynamic is quite different. If one person withdraws, the group lives on. A triad has a different set of relationships. If there are three in a group, two against one dynamics can develop and they exist the potential for majority opinion on the issue. Let's start with the dyads. We get information about dyads from what is called the dyad census, where we calculate the number of mutual, asymmetric and null dyads. What is mutual? Mutual are the dyads with a relationship exists both ways, from node A to node B and back from node B to node A. Asymmetric is where the relationships exist only in one direction. From node A to node B or from node B to node A. Null dyads are the dyads where no relationships exist. Given the certain density one might pause it greater or less than neutral it could based in one relation because when we have a smaller network with a fewer nodes, it's easier to build mutual dyads. What is mutuality? It's the extent to which two actors reciprocate each other's friendship or other interactions. In networks, we calculate mutuality index or M. Test for M compares actual count to expected number of mutuals. We'll talk about what the expected number means. There is a large sample test that assumes approximate normality and we also assume a random graph distribution. We can also include the reciprocity effect in some statistical models. Now, let's talk about triads and transitivity. Triad is a threesome and all the possible ties among them. The two major theoretical concerns are transitivity and cycles. We've talked about transitivity a little bit. It is the central concept in network analysis. Transitivity pertains when all nodes in a triad are connected to each other. If A is tied to B and B is tied to C, then the triad will be transitive. If A is also tied to a C, and intransitive if A and C are not tied. From a theoretical perspective, the expectation is that transitivity will prevail. That a friend of a friend will become a friend. This is framed in network literature as the tendency to close the triad or triad closure and is seen as the feature of structural balance. Open triads with only two edges are labeled structural holes and they thought to be unstable configurations. While this claim about stability of structural holes has been empirically undermined by a famous broad study of 1992. There is still some value in thinking about sociability of a network through the clustering of ties. Let's think about triads from a historical perspective. I think it all starts with Heider and his theory of balance, perhaps earlier. But this is the first known writing where the focus was on individual's perception of social cognitive processes which gave rise to a triad POX person, are the individual and object. The structural balance theory was extended by Cartwright and Harary, focusing on a set of individuals instead of just one individual. Triads are very important. It's the smallest social structure that has the rule character of a society with actors A, B, and C, studying directed triads, they fight tendencies for equilibrium and consistency, also called institutionalization. Triads are also the smallest structures where we can find hierarchy. Triads allow for a much wider range of possible relations. Triadic analysis takes into account all the different combinations of three individuals and examines the interactions between three individuals no matter what other individuals they are connected to. Let's look at this graph, we have four nodes, A, B, C, and D, and we have four triads, A-B-C, A-B-D, B-D-C and A-D-C. Triadic analysis describes directed interactions between three individuals. There are total of 16 different triads as ample described by Wasserman and Faust in 1994. Each triad is represented by three numbers and the letter, if present. First is the number of mutual dyads that are present in the triad, second is the number of asymmetric dyads, and third is the number of null dyads. The acronym is man, so you can always remember that we first specify mutual, then asymmetric, then null. The letter if present of the triad represents a state, whether the relationship goes up, down, whether it's transitive or cycle, and we'll talk about what these letters mean on an actual example. Remember that the number of triads that are present are g23 where g is the number of nodes in the network. In a directed network where we have reciprocated, unreciprocated, and null ties, there are 16 possible triads. Triad census provides us with a frequency of each structure in a network. We can then compare frequencies in our observed networks against deviations, against frequencies in random networks and see how different is our network from random. We'll talk about that when we talk about triads census. For now, let me show you how easy it is to name the triads. Let's go back to our A,B,C,D relationships, and look at the triad A>B>C. We have three connections there, but notice we don't have any mutual dyads, so m is 0. Then we'll have three asymmetric C to A, B to A, and C to B, so we have three asymmetric, and we also have zero null, all connections are present. The numbers are 0, 3, 0, but also notice that C is connected to both A and B, and B is connected to A, and we just talked about the fact that this is the notion of transitivity. Therefore, this is a transitive triad and the letter is T, therefore, the entire structure is 030T. Let's look at the A>B>D triad. We have one null connection between A and D, there is no tie, so it's a one. We have one asymmetric from B to A, and we have one mutual from B to D and D to B, so mutual is one, asymmetric, one, null one, and this is 111. But notice that we could have drawn this two different ways. The phenotype from A to B or B to A. At this point, we have the tie from B to A. This is 111 up, why up and not down. Because when we have a mutual dyad, mutual becomes the base of our triangle, and this as a result, is 111U, B>D>C is another triad, and again, it's very similar to what we've seen before. There is one mutual between B and D, there is one asymmetric between C and B and one null. But now that asymmetric connection goes from C to B, down to the mutual dyad. As a result, we have 111 down. Let's return to letters one more time. We already know that if we have a mutual dyad in a triad, then the mutual becomes the base, as will be the case with 120D and 120U. But what if we do not have a mutual dyad? Look at 021D and 021U. In this case, the structure is the same because null dyad becomes the base. In the case of 021D, the two asymmetric connections go towards the null dyad, and in 021U they go up from the null dyad. Now that we know how to name triads, let's talk about the analysis and what it tells us. What's triadic census?