First, let's start by talking about what these models are, Exponential Random Graph Models. If you are not really big on mathematics, the name might sound scary. Don't worry, you will see that they're actually not that difficult to estimate and in fact, they're very intuitive. Let's try to build the intuition of Exponential Random Graph Models. Exponential Random Graph Models allow us to model social networks parametrically. There's a long research traditions, statistics and the random graph theory that has led to parametric models of networks. These are models of the entire graph. Though, as we'll see, they often work on the dyads in the graph or some other components. Substantively, the approaches to ask whether the graph in question is an element of the class of all random graphs with a given known parameters. For example, imagine we have a graph with five nodes and three edges. We simulate such graphs, and then we'll ask probabilistically, what's the probability of observing the current graph given the conditions of five nodes and three edges. Early p star estimation came from the logic model estimation procedure that was popularized by Wasserman colleagues. There's a very early guide to this approach called a Practical Guide to Feeding p star Models via Logistic Regression. I put it in the materials for this course for your reference, it's actually a fun read. It's amazing how far we have gone since that work was published. The earliest approaches are based on simple random graph theory, and there's been a flurry of activity in the last 30 years or so. The key historical references are Holland and Leinhardt, 1981, Frank and Strauss 1986. We've talked about that already, and of course, Wasserman and Faust,1994, specifically Chapters 15 and 16 of that legendary book, and Wasserman and Pattison in 1996, and they believe that's the article that define the model. The P1 model of Holland and Leinhardt;s is the classic foundation. It lays out the basic idea that you can generate the statistical model of the network by predicting the counts of types of ties asymmetric, null, mutual. They formulate a log-linear model for these counts, and the models is equivalent to a logit model on the dyads. For example, it's the probability of X_ij equal to 1 given some parameters, Alpha plus Beta plus the probability of X_ij. Notice the subscripts. It implies a distinct parameter for every node i and j in the model plus 1 for reciprocity, so it's Alpha i beta j plus the probability of X_ij. Modelling networks parametrically means what? It means we can include parameters. What are those parameters? Well, expansiveness and attractiveness. For example, we can have dummies for each sender and receiver in the network and other parameters and degree distribution. What are the degrees possible? Do we have disconnected nodes with degree zero and do we have nodes with degree that includes almost all other nodes in the network, meaning g minus 1? We can look at mutuality. How many mutual ties do we have? We can look at group membership and by group we can mean anything starting with a triad. We can look at triad transitivity, or intransitivity. That's an important aspect of a tie formation. There are other parameters, some of them we haven't really talked about, but you will encounter them in this course and elsewhere, such as k stars in and out, cyclicity, and then we move on to covariates, meaning the attributes of the node. We have node level covariates such as matching or the difference. For example, when analyzing Florentine families network, we can look at the difference in wealth the canals have. We can look at the edge level of covariates, which is dyad level features such as information processing. For our HR turnover study, we can look at strength of a tie, and of course, we can look at temporal data. There are lots of extensions of ERGM model, not just to temporal data, but [inaudible] taken to account relations in prior waves of network studies. What is the essence of the model? Think of a network as a relation defined on the collection of individuals. We have Mary and we have Paul. Mary relates to Paul, let's say she relates to him in the way of going for an advice. We conceive a network as a relation defined on that advice, or we can say she considers him a friend. That's the second connection. Now, from that connection, we can conceive a graph as the collection of variables X_ij. X_ij is equal to 1, if i makes a tie to j or 0 otherwise, and now this collection of graph is actually translated into our network where we have an on, off switch. On for x equal to 1, off for x equal to zero. Now think of a graph as a collection of tie variables which we can easily transfer into a matrix. First we have a matrix of ties X_ij's, and then we actually have a graph as the collection of those ties. Instead of people connecting to each other, we have nodes i, j, k, l. By consuming it that way, you can see how easily we can move from a relation to a graphical representation to a matrix and the network, and all we have to do at this point is add covariates. What creates heterogeneity in the probability of a tie being formed. Of course, an attribute of a node. We can do the heterogeneity by group for example, average activity or mixing in the group, or individually heterogeneity, men versus women. We can look at the attributes of ties, and that could be heterogeneity and tie duration. Remember, I showed you a survey we did where we asked for the duration of time. Duration of time can become a parameter in the model, and then of course, we can have various configurations of a structure in our network. Notice that the first two attributes of nodes and links are dyad independent. They actually are not related to dyads where the configurations of the network structure, our dyad dependent terms. Therefore, the p star models are hybrid. They're traditional generalized linear models, statistical models. But the unit of analysis is a relation or a dyad, not an individual. Observations may be dependent like an a complex system. Remember that the traditional statistical analysis, we require independence. Here, not only we do not require independence, we actually explicitly model it. We can have complex, non-linear and threshold effects, and the process of estimation, of course, is different. It's a combination of agent-based mathematical models where we can also estimate model parameters from the data, but also estimate the model goodness of fit. Yes, I have not talked about that yet. But it's important to see how good is our model, does it really represent the data that we built it from. The Exponential Random Graph Models has several modelling components. We have a vector of parameters such as regression coefficients. We have a vector of network statistics conditioning the graph itself, and then we'll have a normalizing constant to ensure that all probabilities sum up to 1. That's the basic essence. If we look at that a little bit more detail, we have those vectors, we have the coefficients, let's say homogeneity constraints for coefficients in terms or covariates. What are the terms in the model? Covariates can be edges or wealth or anything else, and coefficients is the probability. Each edge has the same probability versus each edge has a different probability of time formation. We can have some edges with different probabilities. The essence of ERGM model is in the estimation. We estimate the model parameters when we get the output, we will not interpret it directly right away. Because our parameter estimates are not actually estimates we can interpret directly. There are log odds of a dyad formation not even the probabilities, logarithms of the odds and odds if you remember, is the probability of a success divided by the probability of a failure. We have positive or negative parameter estimates if they're statistically significant, positive parameter estimates indicate more configurations observed in the network and expected by chance. Negative parameter estimates indicate fewer configurations that are expected by chance. For example, if we have a positive sign on now triangles, that means that we have more configurations of triangles in our network that could be expected by chance alone. We want to know how the global network structure might have been built out of those small local substructures. The parameter estimates allow us to make inferences about that. Now we're ready to actually look at how Exponential Random Graph Models are built.