  In a seminal paper in 2009, Borcea, Br\"and\'en, and Liggett described the
connection between probability distributions and the geometry of their
generating polynomials. Namely, they characterized that stable generating
polynomials correspond to distributions with the strongest form of negative
dependence. This motivates us to investigate other distributions that can have
this property, and our focus is on random graph models. In this article, we
will lay the groundwork to investigate Markov random graphs, and more generally
exponential random graph models (ERGMs), from this geometric perspective. In
particular, by determining when their corresponding generating polynomials are
either stable and/or Lorentzian. The Lorentzian property was first described in
2020 by Br\"and\'en and Huh and independently by Anari, Oveis-Gharan, and
Vinzant where the latter group called it the completely log-concave property.
The theory of stable polynomials predates this, and is commonly thought of as
the multivariate notion of real-rootedness. Br\"and\'en and Huh proved that
stable polynomials are always Lorentzian. Although it is a strong condition,
verifying stability is not always feasible. We will characterize when certain
classes of Markov random graphs are stable and when they are only Lorentzian.
We then shift our attention to applications of these properties to real-world
networks.
