Directed Networks with a Differentially Private Bi-degree Sequence

TL;DR

This paper introduces a method for releasing differentially private bi-degree sequences of directed networks using Laplace noise, and demonstrates that degree parameter inference remains consistent without denoising, revealing new variance phenomena.

Contribution

It develops a differentially private release mechanism for bi-degree sequences and shows that inference remains valid without denoising, unlike previous assumptions.

Findings

Estimator is asymptotically normal without denoising

Additional variance appears with increased noise

Proposed algorithm efficiently finds the closest graphical bi-degree sequence

Abstract

Although a lot of approaches are developed to release network data with a differentially privacy guarantee, inference using noisy data in many network models is still unknown or not properly explored. In this paper, we release the bi-degree sequences of directed networks using the Laplace mechanism and use the $p_0$ model for inferring the degree parameters. The $p_0$ model is an exponential random graph model with the bi-degree sequence as its exclusively sufficient statistic. We show that the estimator of the parameter without the denoised process is asymptotically consistent and normally distributed. This is contrast sharply with some known results that valid inference such as the existence and consistency of the estimator needs the denoised process. Along the way, a new phenomenon is revealed in which an additional variance factor appears in the asymptotic variance of the estimator when the noise becomes large. Further, we propose an efficient algorithm for finding the closet point lying in the set of all graphical bi-degree sequences under the global $L_1$ optimization problem. Numerical studies demonstrate our theoretical findings.