Algebraic statistics for a directed random graph model with reciprocation

TL;DR

This paper applies algebraic statistics to analyze the p_1 directed graph model with reciprocation, revealing its toric structure and studying Markov bases for improved estimation and testing in social network analysis.

Contribution

It demonstrates that the p_1 model is a multi-homogeneous toric model and provides an extensive study of its Markov bases, advancing algebraic methods in social network modeling.

Findings

p_1 model is a multi-homogeneous toric model

Characterization of Markov bases for p_1 models

Implications for estimation and goodness-of-fit testing

Abstract

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.