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

Releasing the exact degree sequence of a graph for analysis may violate privacy. However, the degree sequence of a graph is an important summary statistic that is used in many statistical models. Hence a natural starting point is to release a private version of the degree sequence. A graphical degree partition is a monotonic degree sequence such that there exists a simple graph realizing the sequence. Ensuring graphicalness of the released degree partition is a desirable property for many statistical inference procedures. We present an algorithm to release a graphical degree partition of a graph under the framework of differential privacy. Unlike previous algorithms, our algorithm allows an analyst to perform meaningful statistical inference from the released degree partition. We focus on the statistical inference tasks of existence of maximum likelihood estimates, parameter estimation and goodness of fit testing for the random graph model where the degree partition is a sufficient statistic, called the beta model. We show the usefulness of our algorithm for performing statistical inference for the beta model by evaluating it empirically on simulated and real datasets. As the degree partition is graphical, our algorithm can also be used to release synthetic graphs.