  In this paper, we explore the two-star Exponential Random Graph Model, which
is a two parameter exponential family on the space of simple labeled graphs. We
introduce auxiliary variables to express the two-star model as a mixture of the
$\beta$ model on networks. Using this representation, we study asymptotic
distribution of the number of edges, and the sampling variance of the degrees.
In particular, the limiting distribution for the number of edges has similar
phase transition behavior to that of the magnetization in the Curie-Weiss Ising
model of Statistical Physics. Using this, we show existence of consistent
estimates for both parameters in all parameter domains. Finally, we prove that
the centered partial sum of degrees converges as a process to a Brownian bridge
in all parameter domains, irrespective of the phase transition.
