The Annals of Statistics

Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

Ting Yan, Chenlei Leng, and Ji Zhu

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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.

Article information

Ann. Statist., Volume 44, Number 1 (2016), 31-57.

Received: December 2014
Revised: May 2015
First available in Project Euclid: 10 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation 62F12: Asymptotic properties of estimators
Secondary: 62B05: Sufficient statistics and fields 62E20: Asymptotic distribution theory 05C80: Random graphs [See also 60B20]

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


Yan, Ting; Leng, Chenlei; Zhu, Ji. Asymptotics in directed exponential random graph models with an increasing bi-degree sequence. Ann. Statist. 44 (2016), no. 1, 31--57. doi:10.1214/15-AOS1343.

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Supplemental materials

  • Supplement to “Asymptotics in directed exponential random graph models with an increasing bi-degree sequence.”. The supplemental material contains proofs for the lemmas in Section 2.2, the theorems and lemmas in Sections 2.3 and 2.4, Proposition 1 and Theorem 7.