Next, we'll turn our attention to tie formation. Why do ties form? As we have seen already in this course, ties can form randomly. For example, whether two children in the first grade become friends may very much become the chance of them being assigned to the same teacher. Otherwise, they may never meet in a large school. Assignment of children to grades is random. Therefore, two children becoming friends is a function of a random process. What role do random processes play in tie formation? Let's talk about that. There are many processes that underlie tie formation. For example, a study by Mcpherson, Smith-Lovin, and Cook 2001, identifies homophily, shared affiliations, spatial propinquity are the processes that matter in the exogenous effects. Other studies, for example, by Feld 1981, point to foci for network tie formation or settings, as White in 1995 or Pattison and Robinson 2002 indicate. There are also endogenous network effects such as clustering. Tie formation is often more likely when actors have network partners in common, and there are lots of studies have come from that. For example, Cartwright and Harary in 1956 or Granovetter 1973. I'm sure you've all have heard of Strength of Weak Ties. But more generally, a social tie's subject to and known to be subject to the hegemonic pressures of others engaged in social construction of the network, as White said in 1998. There are also interactions between exogenous and endogenous effects, and that is called the general social selection. Refer to Robins, Elliott, and Pattison of 2001 if you want to learn more. But what we need to do is we need to model endogenous and exogenous network principles. Let's start with endogenous. The guiding principles are that the network ties are the outcomes of observed social processes that tend to be local and interactive. They are both irregularities and irregularities in those local interactive processes. What does that mean? That means that we have to look at the network on the local level and examine the structure of this network to see if the structure of the network is responsible for the tie formation. Maybe we need to examine attributes of actors on the local level, but that's still inside the network, the endogenous local level. We construct statistical models in which local interactivity is permitted, and assumptions about form of local interactions are explicit. Regularities are represented by model parameters, and they're estimated from the data. For example, we can look at how many transitive triads we have in the network and try to model that parameter as the predictor of tie formation. We can also look at consequences of local irregularities for global network. Do the processes that apply at the local level also apply to a global level? We can also talk about properties that can be understood and can provide us with an exact approach to model evaluation. For example, if there is a tendency for mutuality, can we use this mutuality index to evaluate the quality of the model that we have built. Think about local interactivity. We model tie variables X_ij. X_ij is equal to one if I has a tie to a j. Otherwise, it's equal to zero. The realization of that X is noted by X_ij. We can also incorporate node-level exogenous attribute variables Y. We have two types of information that we can use to build our models. First, the ties themselves and all the structures that those ties form. But also, we can incorporate the attribute information, which is characteristics of the nodes that are parts of those ties, and we can observe whether or not certain characteristics are conducive to more tie formation. There are two modeling steps in general. First, we define two network tie variables to be neighbors if they are conditionally dependent, given the value of all other tie variables. That's easy. But then what are the appropriate neighbor assumptions? How do we define neighbors? We have to come up with some dependency assumptions. The need for dependency assumptions gets us to those random graphs that I've talked about. The simple dyadic dependence hypothesis is the Bernoulli random graphs. What's Bernoulli? Well, it's like flipping a coin. We flip a coin, we get either heads or tails. There are no neighborhood relations if all edges are independent. That's what Bernoulli random graph and the two tie variables and neighbors if they're in the same dyad. Well, hopefully, if real-life need something more realistic than Bernoulli because we don't build relationships based on the flip of a coin. Let's try a Markov model. Two tie variables are neighbors if they share an actor. For example, that's Frankenstein's suggested in 1986. Or they can share connections, which means it's idealization dependent model with two existing ties. Meaning completing a social circle. There are other possibilities, but just these two can get us a long way. There are some models for interactive systems of variables and will define the neighborhood as a set of mutually neighboring variables that corresponds to a potential network configuration. For example, X_12, X_13, and X_23 can correspond to a triangle of the triad that we're already familiar with. Now Hammersley-Clifford theorem states that the model for X has a form determined by its neighborhoods. This general approach leads to exponential random graph or p-star models. You can look at Frank and Strauss 1986, extended by Wasserman, Pattison, and Robins. With these tie formation processes, be they random or explained with help of probability theory, we can now move to social processes in general and talk about those random graphs.[MUSIC]