Near-Optimal Differentially Private k-Core Decomposition
CoRR(2023)
摘要
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.
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