  Holland and Leinhardt (1981) proposed a directed random graph model, the p1
model, to describe dyadic interactions in a social network. In previous work
(Petrovic et al., 2010), we studied the algebraic properties of the p1 model
and showed that it is a toric model specified by a multi-homogeneous ideal. We
conducted an extensive study of the Markov bases for p1 that incorporate
explicitly the constraint arising from multi-homogeneity. Here we consider the
properties of the corresponding toric variety and relate them to the conditions
for the existence of the maximum likelihood and extended maximum likelihood
estimators or the model parameters. Our results are directly relevant to the
estimation and conditional goodness-of-fit testing problems in p1 models.
