Algebraic statistics for random graph models: Markov bases and their uses
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We use algebraic geometry to study a statistical model for the analysis of networks represented by graphs with directed edges due to Holland and Leinhardt, known as p1, which allows for differential attraction (popularity) and expansiveness, as well as an additional effect due to reciprocation. In particular, we attempt to derive Markov bases for p1 and to link these to the results on Markov bases for working with log-linear models for contingency tables. Because of the contingency table representation for p1 we expect some form of congruence. Markov bases and related algebraic geometry notions are useful for at least two statistical problems: (i) determining condition for the existence of maximum likelihood estimates, and (ii) using them to traverse conditional (given minimal sufficient statistics) sample spaces, and thus generating ``exact'' distributions useful for assessing goodness of fit. We outline some of these potential uses for the algebraic representation of p1.
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