### Parameterizing Exponential Family Models for Random Graphs:
Current Methods and New Directions

### Speaker: Carter Butts

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 θ^{T} 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.