Many deployments of differential privacy in industry are in the local model, where each party releases its private information via a differentially private randomizer. We study triangle counting in the noninteractive and interactive local model with edge differential privacy (that, intuitively, requires that the outputs of the algorithm on graphs that differ in one edge be indistinguishable). In this model, each party's local view consists of the adjacency list of one vertex. In the noninteractive model, we prove that additive $\Omega(n^2)$ error is necessary, where $n$ is the number of nodes. This lower bound is our main technical contribution. It uses a reconstruction attack with a new class of linear queries and a novel mix-and-match strategy of running the local randomizers with different completions of their adjacency lists. It matches the additive error of the algorithm based on Randomized Response, proposed by Imola, Murakami and Chaudhuri (USENIX2021) and analyzed by Imola, Murakami and Chaudhuri (CCS2022) for constant $\varepsilon$. We use a different postprocessing of Randomized Response and provide tight bounds on the variance of the resulting algorithm. In the interactive setting, we prove a lower bound of $\Omega(n^{3/2})$ on the additive error. Previously, no hardness results were known for interactive, edge-private algorithms in the local model, except for those that follow trivially from the results for the central model. Our work significantly improves on the state of the art in differentially private graph analysis in the local model.

翻译：差分隐私在工业界的许多部署采用局部模型，其中每个参与方通过差分隐私随机化器发布其私有信息。我们研究非交互式和交互式局部模型中具有边差分隐私（直观上要求算法在图仅相差一条边时的输出不可区分）的三角形计数问题。在该模型中，每个参与方的局部视图由单个顶点的邻接列表组成。在非交互式模型中，我们证明加法误差至少为$\Omega(n^2)$，其中$n$为节点数。该下界是我们的主要技术贡献，它采用包含新型线性查询的重构攻击，以及一种新颖的混合匹配策略，通过使用邻接列表的不同补全形式运行局部随机化器。这一结果匹配了Imola、Murakami和Chaudhuri（USENIX 2021）提出、并由Imola、Murakami和Chaudhuri（CCS 2022）针对常数$\varepsilon$分析的基于随机响应的算法的加法误差。我们采用不同的随机响应后处理方法，并给出由此产生算法方差的紧界。在交互式设置中，我们证明了加法误差的$\Omega(n^{3/2})$下界。此前，局部模型中交互式边隐私算法（除中心模型结果平凡导出的结论外）未有任何硬度结果。本工作显著改进了局部模型中差分隐私图分析的最新进展。