Recursive mechanism: towards node differential privacy and unrestricted joins

Existing differential privacy (DP) studies mainly consider aggregation on data sets where each entry corresponds to a particular participant to be protected. In many situations, a user may pose a relational algebra query on a database with sensitive data, and desire differentially private aggregation on the result of the query. However, no existing work is able to release such aggregation when the query contains unrestricted join operations. This severely limits the applications of existing DP techniques because many data analysis tasks require unrestricted joins. One example is subgraph counting on a graph. Furthermore, existing methods for differentially private subgraph counting support only edge DP and are subject to very simple subgraphs. Until recent, whether any nontrivial graph statistics can be released with reasonable accuracy for arbitrary kind of input graphs under node DP was still an open problem. In this paper, we propose a novel differentially private mechanism that supports unrestricted joins, to release an approximation of a linear statistic of the result of some positive relational algebra calculation over a sensitive database. The error bound of the approximate answer is roughly proportional to the empirical sensitivity of the query --- a new notion that measures the maximum possible change to the query answer when a participant withdraws its data from the sensitive database. For subgraph counting, our mechanism provides a solution to achieve node DP, for any kind of subgraphs.