Inference in the $p_0$ model for directed networks under local differential privacy

TL;DR

This paper develops a method for estimating parameters in directed network models under local differential privacy by using noisy bi-degree sequences, ensuring asymptotic consistency and normality, and compares different perturbation mechanisms.

Contribution

It introduces a novel approach using edge-flipping noise to estimate $p_0$ model parameters from privatized network data, with theoretical guarantees and practical evaluation.

Findings

Private estimators are asymptotically consistent and normally distributed.

Output perturbation can outperform input perturbation in estimation accuracy.

Numerical studies validate theoretical results and demonstrate practical effectiveness.

Abstract

We explore the edge-flipping mechanism, a type of input perturbation, to release the directed graph under edge-local differential privacy. By using the noisy bi-degree sequence from the output graph, we construct the moment equations to estimate the unknown parameters in the $p_0$ model, which is an exponential family distribution with the bi-degree sequence as the natural sufficient statistic. We show that the resulting private estimator is asymptotically consistent and normally distributed under some conditions. In addition, we compare the performance of input and output perturbation mechanisms for releasing bi-degree sequences in terms of parameter estimation accuracy and privacy protection. Numerical studies demonstrate our theoretical findings and compare the performance of the private estimates obtained by different types of perturbation methods. We apply the proposed method to analyze the UC Irvine message network.