DIRECTED NETWORKS WITH A DIFFERENTIALLY PRIVATE BI-DEGREE SEQUENCE

Although many approaches have been developed for releasing network data with a differential privacy guarantee, few studies have examined inferences in network models with differential privacy data. Here, we propose releasing bi-degree sequences of directed networks using the Laplace mechanism and making inferences using the p(0) model, which is an exponential random graph model with the bi-degree sequence as its exclusively sufficient statistic. We show that the estimator of the parameters without the so-called denoised process is asymptotically consistent and normally distributed. This is in sharp contrast to some known results that valid inferences (e.g., the existence and consistency) of an estimator require denoising. We also show a new phenomenon, in which an additional variance factor appears in the asymptotic variance of the estimator to account for the noise. An efficient algorithm is proposed for finding the closest point in the set of all graphical bi-degree sequences under the global L-1-optimization problem. A numerical study demonstrates our theoretical findings.