Let's talk about roles and positions which help us bring the mathematics of log modeling to social networks. For example, consider this study by wide at all, from logical role systems to empirical social structures. Let's start with a basic idea of what the role is. It doesn't exchange of something such as support the ideas commands between the actors. But those actors occupy different roles. In this graph. We have a husband and a wife, and then we'll have their three children. Notice how the rules are different between members of this network. Husband and wives are spouses to each other, their parents to their children, and then the children or siblings to each other and children to their parents. Now, of course, there are many other relations within a family, but we can actually specify things like romantic love, which exists only between the husband and the wife, nodes, or bickering which may exist between parents, but much more likely to exist between children. In those roles that we specify, we can actually identify all families that have similar structures. In the larger community can create a block model based on these roles. Roles and positions are defined relationally. Position is a collection of actors correlate in similar ways to actors in other positions. The role is the relational pattern, typically linking those positions, which brings us to the next set of definitions. The first one is the equivalence. Actors in the same position are equivalent if their ties to and from others are the same. Think of a husband and a wife. If we name them Spouse 1 and Spouse 2, from their relations to each other and to all of their children, we would be able to tell which one is which if you don't know the names. That means they are equivalent and we can tell the difference between them. Blocks. A region of the graph that shows the relations of two groups of actors with equivalent roles or positions. If there are key positions then there will be key by key blocks. If we have four positions, we'll have a block model that has the structure of four-by-four. What is a block model? It does it reduced form of a network. Actors are partitioned into positions and ties between positions are represented somehow. Let's think about representing relations measured in actors and the compound relations constructed from them. We can study patterns of interdependence of relations. That would allow us to build models of social roles and role structures. Positions in block modeling are also great for studying patterns of dependencies among different types of ties or relationship. It's ideal for more relational, binary, or dichotomous networks, of course, with some limitations. It's not really a perfect methodology, but some researchers have claimed it to be the perfect methodology for the presentations of global little structures where the position is a collection of individuals embedded in the natural correlations. That all is patterns of relations which obtained among actors or among politicians. Actors in the position are similar in their social or behavioral activity and ties or interactions with respect to actors in other positions, so all parents appearance to their children. There are two general approaches to building block models. The first one is the indirect approach, where we measure the degree of equivalents for peers of actors, then we specify the equivalence, identify positions of equivalent actors, and then the representatives between positions. Direct approach is a little different. We specify the equivalence and then identify partition of actors that is optimal for that specified measure. Actors in the same position are equivalent in their ties to and from others. There are several ways to identify those equivalences. In general, there are three definitions. Structural, automorphic, and regular. Structural is the strictest. Actors are structurally equivalent if they connect to the rest of the network in exactly the same way, which means that structurally equivalent actors have identical relational ties to and from all other actors in a network. Automorphic equivalence is little less strict. Actors automorphically equivalent if then when you remove their labels they're structurally indistinguishable from one another. That's an example of a husband and a wife I have given. You can't distinguish them once you remove their labels from the relations to their children. The regular equivalence is actors are regular cool if they have identical ties to and from equivalent others. That's the least strict definition of equivalence. To put it in a more intuitive way, equivalent units have the same connections to the same actress for the structural, have the same or similar patterns to possibly different actors is the regular. It is important to note that this type of analysis is largely concerned with the connection that two [inaudible] actors might have with one another. Consider this network. We have eight nodes and we have very different positions. For structural column to have six. H and G are structurally equivalent to each other. They're connected to F exactly the same way. E and D are structural equivalent. They're connected to C exactly the same way. Then A, B, C, and D create separate sets. There's actually only one unit in the set because they're not structurally equivalent to anyone else. For automorphically equivalence, we have four positions. H, G, E, and D are automorphically equivalent because they're connected in a similar way to nodes F and G. But then that becomes also true for this set F C, which is automorphically equivalent to nodes B, and then H, G for F, E and D for C. We have two sets with just one nodes in them, A and B. For regular equivalent positions, we actually have two sets, A H G E D, and B F C. Try to figure out why that is. There's an alternative notion of equivalence instead of the exact same ties to exact same alters, which is very difficult to find in a large network. We can look for nodes with similar ties to similar types of alters. If we have school children and their parents, we can look to those similar ties because we can identify parents as one group and children as another group. While most of the colons measures focus on the position within the full network, some measures focus only on the patterns within the local Thai neighborhood. Note that structural equivalent actors are always automorphically equivalent. Automorphically equivalent actors are regular equivalent. Structurally equivalent and automorphically equivalent actors are role equivalent. In practice, we tend to ignore some of these fine distinctions as they are blurred quickly, once we have to operationalize them on the real graph. It turns out that few people are ever exactly equivalent and we'll have to approximate the links between the types. In all cases, the procedure can work over multiple relations simultaneously. This process of identifying position is called block modeling, and requires identifying a measure of similarity among nodes. Then you know the rest. We can measure equivalence in reality with real data. Actors since they are never perfectly structurally equivalent, can occupy similar roles and positions. We can locate subsets of actors that are approximately structurally equivalent or that occupies similar roles. Then we'll compare entries in the rows and columns of sociomatrices to do so. Of course, it's all done by a program nowadays, not by hand. The way that we establish those blocks is we create the similarity or dissimilarity measure for all actors using different methods such as correlation, particularly Pearson, Euclidean distance or something else. Then we identify patterns with cluster analysis, multidimensional scaling factor analysis. There is a special method called CONCOR, which is iteration of conversion correlations. Then we can do a direct search and for is perfect as possible of a structure, and that's called a Tabu search. The steps for positional analysis are as follows, specify a formal definition of equivalence. Specify a measure of the degree to which sets of actors approach that definition in a given set of network data. Represent the equivalences and assess the adequacy of the presentation. Recent work has generalized log models into direction. The specific structural hypothesis, for example, Core-periphery models, or generalized block modeling based on particular relationship types. Indirect or exploratory, which is also called conventional block modeling, is used quite a bit. It does not deal with data directly. It transforms and clusters the data, such as with dissimilarity measures and cluster analytic methods. It restrict blocks to null, regular or complete. These blocks can appear anywhere in the block model. Measure of fit are quite rare. The direct or confirmatory generalized block modeling doesn't deal with data directly and allows for deductive approaches. It expands the types of blocks available to fit and substantive concern about how they are related. It does allow for an assessment of fit. Which one should you use? It all depends on the question. I'll show you both approaches in our lecture, which we are actually ready to move to already. Are you ready? I am. Let's go.