  The p_1 model is a directed random graph model used to describe dyadic
interactions in a social network in terms of effects due to differential
attraction (popularity) and expansiveness, as well as an additional effect due
to reciprocation. In this article we carry out an algebraic statistics analysis
of this model. We show that the p_1 model is a toric model specified by a
multi-homogeneous ideal. We conduct an extensive study of the Markov bases for
p_1 models that incorporate explicitly the constraint arising from
multi-homogeneity. Our results are directly relevant to the estimation and
conditional goodness-of-fit testing problems in p_1 models.
