Let G be a random graph (typically on some finite support S), such that the log-probability of a given realization G=g may be written as the inner product θT t(g) - ψ(θ,S), where θ is a constant real vector of length k, t is a k-vector of real-valued functions on S, and ψ is a real-valued function depending only on θ and S. The distribution of G written in this way is said to be in exponential family form, and the vector t defines a discrete exponential family of distributions on S. Such exponential families are increasingly used as models for the structure of social networks, since they provide a highly general mechanism for parameterizing dependence among edges and are supported by a growing body of computational and inferential theory. An important and ongoing challenge within this line of research is to find principled ways of proposing vectors of graph statistics (i.e., t) which parsimoniously capture the dependence associated with particular social mechanisms. Here, I will review some current approaches to the parameterization problem, as well as an alternative based on the use of potential games. I will briefly illustrate the application of these techniques to the modeling of interpersonal relationships, and show how the respective parameterization schemes can aid in the interpretation of both new and existing models for relational data.