This chapter is written for those interested in a more detailed understanding of how assumptions regarding various forms of dependence can be formalized. What is treated here is not essential for applying exponential random graph models (ERGMs) and may be skipped at a first reading. The general idea, laid down by Frank and Strauss (1986), is nevertheless crucial to the formulation of statistical models treated in this book.

Important key points in this chapter are as follows:

• Subgraph counts are not arbitrarily chosen in ERGMs but correspond to specific dependency structures.

• The subgraph counts in ERGMs are intricately nested and interdependent, so care has to be taken in interpreting parameters in isolation.

• An ERGM is akin to a log-linear model where the subgraph counts are represented by interactions of tie-variables.

• ERGMs try to reduce the complexity of observed networks into systematic underlying principles and stochastic components.

• A homogeneous ERGM assigns equal probability to graphs that are structurally identical.

In this chapter, we focus on models for undirected graphs. Dependence graphs for directed models are a natural extension of what we describe here, but we only discuss them briefly.

Figure 7.1. Tie-variables of (a) four-node graph and (b) associated Bernoulli dependence graph.