Statistics and Its Interface

Volume 8 (2015)

Number 3

A simulation study of the $p_1$ model for directed random graphs

Pages: 255 – 266



Ting Yan (Department of Statistics, Central China Normal University, Wuhan, China)

Chenlei Leng (Department of Statistics, University of Warwick, United Kingdom)


The $p_1$ exponential-family distribution in Holland and Leinhardt (1981) is one of the earliest models for directed random graphs and has been widely used in practice. The conditions for the existence and uniqueness of the maximum likelihood estimate (MLE) have been derived. However, it is a daunting task to investigate the large-sample properties of the MLE theoretically as the number of graphical vertices goes to infinity. The uniform consistency and asymptotic normality of the MLE have been derived for some specialized models closely related to the $p_1$ model but general results are lacking. In this article, we explore the consistency and asymptotic normality of the MLE in the $p_1$ model as the network size increases through numerical simulations. The results indicate that the $p_1$ model also enjoys good asymptotic properties.


directed random graphs, increasing number of parameters, maximum likelihood estimation, $p_1$ exponential-family distribution

2010 Mathematics Subject Classification

Primary 62F12. Secondary 05C80, 62E20, 62F10.

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Published 17 April 2015