The Annals of Statistics

Inference using noisy degrees: Differentially private $\beta$-model and synthetic graphs

Vishesh Karwa and Aleksandra Slavković

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The $\beta$-model of random graphs is an exponential family model with the degree sequence as a sufficient statistic. In this paper, we contribute three key results. First, we characterize conditions that lead to a quadratic time algorithm to check for the existence of MLE of the $\beta$-model, and show that the MLE never exists for the degree partition $\beta$-model. Second, motivated by privacy problems with network data, we derive a differentially private estimator of the parameters of $\beta$-model, and show it is consistent and asymptotically normally distributed—it achieves the same rate of convergence as the nonprivate estimator. We present an efficient algorithm for the private estimator that can be used to release synthetic graphs. Our techniques can also be used to release degree distributions and degree partitions accurately and privately, and to perform inference from noisy degrees arising from contexts other than privacy. We evaluate the proposed estimator on real graphs and compare it with a current algorithm for releasing degree distributions and find that it does significantly better. Finally, our paper addresses shortcomings of current approaches to a fundamental problem of how to perform valid statistical inference from data released by privacy mechanisms, and lays a foundational groundwork on how to achieve optimal and private statistical inference in a principled manner by modeling the privacy mechanism; these principles should be applicable to a class of models beyond the $\beta$-model.

Article information

Ann. Statist. Volume 44, Number 1 (2016), 87-112.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 91D30: Social networks
Secondary: 62F30: Inference under constraints

Degree sequence differential privacy $\beta$-model existence of MLE measurement error


Karwa, Vishesh; Slavković, Aleksandra. Inference using noisy degrees: Differentially private $\beta$-model and synthetic graphs. Ann. Statist. 44 (2016), no. 1, 87--112. doi:10.1214/15-AOS1358.

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Supplemental materials

  • Supplement to “Inference using noisy degrees: Differentially Private $\beta$-model and synthetic graphs”. This supplementary material contains the proof of the key Theorems 2, 3 and 4 from the paper.