On the Existence of the MLE for a Directed Random Graph Network Model with Reciprocation

A. Rinaldo; S. E. Fienberg; S. Petrovi\'c

On the Existence of the MLE for a Directed Random Graph Network Model with Reciprocation

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arxiv 1010.0745 v1 pith:JBOFZYOB submitted 2010-10-05 math.ST stat.TH

A. Rinaldo , S. Petrović , S. E. Fienberg This is my paper · ORCID

classification math.STstat.TH

keywords modeldirectedexistencegraphlikelihoodmaximumnetworkproperties

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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.

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Modelling Directed Networks with Reciprocity
stat.ME 2024-11 conditional novelty 6.0 of 10

In sparse directed networks, reciprocity is estimated with an effective sample size set by the slower of the two link-count scales, yielding a phase transition in the MLE's limiting distribution.

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