Open Access
February 2016 Inference using noisy degrees: Differentially private $\beta$-model and synthetic graphs
Vishesh Karwa, Aleksandra Slavković
Ann. Statist. 44(1): 87-112 (February 2016). DOI: 10.1214/15-AOS1358


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.


Download Citation

Vishesh Karwa. Aleksandra Slavković. "Inference using noisy degrees: Differentially private $\beta$-model and synthetic graphs." Ann. Statist. 44 (1) 87 - 112, February 2016.


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

zbMATH: 1331.62114
MathSciNet: MR3449763
Digital Object Identifier: 10.1214/15-AOS1358

Primary: 62F12 , 91D30
Secondary: 62F30

Keywords: $\beta$-model , degree sequence , differential privacy , existence of MLE , measurement error

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 1 • February 2016
Back to Top