Near-Optimal Differentially Private k-Core Decomposition

Recent work by Dhulipala, Liu, Raskhodnikova, Shi, Shun, and Yu~\cite{DLRSSY22} initiated the study of the $k$-core decomposition problem under differential privacy. They show that approximate $k$-core numbers can be output while guaranteeing differential privacy, while only incurring a multiplicative error of $(2 +\eta)$ (for any constant $\eta >0$) and additive error of $\poly(\log(n))/\eps$. In this paper, we revisit this problem. Our main result is an $\eps$-edge differentially private algorithm for $k$-core decomposition which outputs the core numbers with no multiplicative error and $O(\text{log}(n)/\eps)$ additive error. This improves upon previous work by a factor of 2 in the multiplicative error, while giving near-optimal additive error. With a little additional work, this implies improved algorithms for densest subgraph and low out-degree ordering under differential privacy. For low out-degree ordering, we give an $\eps$-edge differentially private algorithm which outputs an implicit orientation such that the out-degree of each vertex is at most $d+O(\log{n}/{\eps})$, where $d$ is the degeneracy of the graph. This improves upon the best known guarantees for the problem by a factor of $4$ and gives near-optimal additive error. For densest subgraph, we give an $\eps$-edge differentially private algorithm outputting a subset of nodes that induces a subgraph of density at least ${D^*}/{2}-O(\text{log}(n)/\eps)$, where $D^*$ is the density for the optimal subgraph.