  Sampling algorithms, hypergraph degree sequences, and polytopes play a
crucial role in statistical analysis of network data. This article offers a
brief overview of open problems in this area of discrete mathematics from the
point of view of a particular family of statistical models for networks called
exponential random graph models. The problems and underlying constructions are
also related to well-known concepts in commutative algebra and graph-theoretic
concepts in computer science. We outline a few lines of recent work that
highlight the natural connection between these fields and unify them into some
open problems. While these problems are often relevant in discrete mathematics
in their own right, the emphasis here is on statistical relevance with the hope
that these lines of research do not remain disjoint. Suggested specific open
problems and general research questions should advance algebraic statistics
theory as well as applied statistical tools for rigorous statistical analysis
of networks.
