# Section I Rationale

## What Are Exponential Random Graph Models?

Garry Robins and Dean Lusher

### Exponential Random Graph Models: A Short Definition

Exponential random graph models (ERGMs) are statistical models for network structure, permitting inferences about how network ties are patterned. Put another way, ERGMs are tie-based models for understanding how and why social network ties arise. This focus aligns ERGMs with a principal goal of much empirical social network research, which is to understand a given “observed” network structure (i.e., a network on which a researcher has collected data), and so to obtain insight into the underlying processes that create and sustain the network-based social system.

Much of social network analysis has been concerned with representing the network, a graph *G*, through various summary measures. From the literature, the reader may be familiar with summary measures *z*(*G*) such as the number of edges in *G*, the number of mutual ties, centrality measures, triad census, and so on. We call these summary measures “network statistics,” and in mathematical terms, the ERGM assigns probability to graphs according to these statistics:

Pq (g) = *ce°1 ^{z}1^{(G)}+°2*

^{z}2

^{(G)}+—+

^{q}p

^{z}p

^{(G)}

The probability of a given network *G* is given by a sum of network statistics (the *z*s in this expression) weighted, just as in a regression, by parameters (the *Q*s) inside an exponential (and where *c* is a normalizing constant). The network statistics are counts of the number of network configurations in the given network G, or some function of those counts. These configurations are small, local subgraphs in the network. In short, the probability of the network depends on how many of those configurations are present, and the parameters inform us of the importance of each configuration.

This expression is explained in much more detail in Section II. However, because the mathematical features are not important for our purposes here, we hope to explain ERGMs in a relatively intuitive way in this introductory section.

To put it as simply as possible, a researcher specifies an ERGM by choosing a set of configurations of theoretical interest. As we will see, there are many sets of plausible configurations that can be used. Then, by applying this particular model to an observed social network, parameters are estimated. This permits inferences about the configurations - the network patterns - in the data, and this in turn allows inferences about the type of social processes that are important in creating and sustaining the network. Thus, ERGMs provide a methodology to investigate network structures and processes empirically.

Note that there is not just one ERGM - there are whole classes of them. The researcher has to choose the specification of an ERGM for the data (just as a researcher has to choose the variables to include in a regression). For an ERGM, the specifications involve choices of configurations that the researcher believes are relevant to the network structure. Although there are some standard ways to do this, the choices are ultimately based on theories about how ties come into being and appear in regular patterns. We discuss some of these theories in greater length in Chapter 3. However, an ERGM itself carries some metatheory about networks, a conceptualization of a social network, and how it is created.