# Simplified Account of an Exponential Random Graph Model as a Statistical Model

Garry Robins and Dean Lusher

This chapter provides a simplified overview of some methodological aspects of exponential random graph models (ERGMs), with the technical detail presented in Section II, specifically in Chapters 6 and 7. To begin, it is worthwhile to consider the value of a statistical model in understanding social network structure.

Harrison White made the important observation that “sociology has to account for chaos and normality together” (2008, 1). Social life is stochastic, and social networks are not predetermined or invariant. We do not expect that in a human social network, reciprocity will apply (strictly) in all situations; rather, there may be a tendency toward reciprocity in the sense that more reciprocation will be present than otherwise expected over and above what would result from other processes. In a sense, if we do not allow for “tendencies” with some variation, in the extreme, a nonstochastic model requires one unique explanation for each tie, present or absent.

Accordingly, it makes sense to use a statistical model such as an ERGM to investigate network structure. By incorporating randomness, statistical models deal with expected values, so we are then able to draw inferences about whether observed data are consistent with expectations.

The balance between randomness and order is an important issue in much social network research. For instance, in considering the “small world” nature of many social networks, Watts (1999) showed thatadding a small amount of randomness to a highly structured graph could dramatically shorten path lengths. In an ERGM context, a configuration represents the ordered nature of local structure. If effects for configurations in a model are minimal (e.g., a tendency for reciprocity might be weak), then the resulting networks will be close to purely random. In contrast, if an effect is strong (e.g., a strong tendency for reciprocity), then the resulting networks will appear as highly structured (e.g., most ties will be reciprocated). Because an ERGM is stochastic, the model does

Figure 4.1. (a) Simple random network and (b) empirical communication network.

not imply just one network. The result is a “probability distribution of graphs,” which we discuss in more detail later in this chapter.