Algebraic Statistics for a Directed Random Graph Model with Reciprocation

The p1 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 p1 model is a toric model specified by a multi-homogeneous ideal. We conduct an extensive study of the Markov bases for p1 models that incorporate explicitly the constraint arising from multi-homogeneity. We consider the properties of the corresponding toric variety and relate them to the conditions for existence of the maximum likelihood and extended maximum likelihood estimator. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in pmodels.