Open Access
February 2016 Asymptotics in directed exponential random graph models with an increasing bi-degree sequence
Ting Yan, Chenlei Leng, Ji Zhu
Ann. Statist. 44(1): 31-57 (February 2016). DOI: 10.1214/15-AOS1343


Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.


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Ting Yan. Chenlei Leng. Ji Zhu. "Asymptotics in directed exponential random graph models with an increasing bi-degree sequence." Ann. Statist. 44 (1) 31 - 57, February 2016.


Received: 1 December 2014; Revised: 1 May 2015; Published: February 2016
First available in Project Euclid: 10 December 2015

zbMATH: 1331.62110
MathSciNet: MR3449761
Digital Object Identifier: 10.1214/15-AOS1343

Primary: 62F10 , 62F12
Secondary: 05C80 , 62B05 , 62E20

Keywords: Bi-degree sequence , central limit theorem , consistency , directed exponential random graph models , Fisher information matrix , maximum likelihood estimation

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 1 • February 2016
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