Differentially private algorithms protect individuals in data analysis scenarios by ensuring that there is only a weak correlation between the existence of the user in the data and the result of the analysis. Dynamic graph algorithms maintain the solution to a problem (e.g., a matching) on an evolving input, i.e., a graph where nodes or edges are inserted or deleted over time. They output the value of the solution after each update operation, i.e., continuously. We study (event-level and user-level) differentially private algorithms for graph problems under continual observation, i.e., differentially private dynamic graph algorithms. We present event-level private algorithms for partially dynamic counting-based problems such as triangle count that improve the additive error by a polynomial factor (in the length $T$ of the update sequence) on the state of the art, resulting in the first algorithms with additive error polylogarithmic in $T$. We also give $\varepsilon$-differentially private and partially dynamic algorithms for minimum spanning tree, minimum cut, densest subgraph, and maximum matching. The additive error of our improved MST algorithm is $O(W \log^{3/2}T / \varepsilon)$, where $W$ is the maximum weight of any edge, which, as we show, is tight up to a $(\sqrt{\log T} / \varepsilon)$-factor. For the other problems, we present a partially-dynamic algorithm with multiplicative error $(1+\beta)$ for any constant $\beta > 0$ and additive error $O(W \log(nW) \log(T) / (\varepsilon \beta) )$. Finally, we show that the additive error for a broad class of dynamic graph algorithms with user-level privacy must be linear in the value of the output solution's range.
翻译:不同的私人算法通过确保数据用户的存在与分析结果之间仅有薄弱的关联性,在数据分析情景中保护个人。 动态图形算法维持对不断演变的输入中的问题( 例如匹配) 的解决方案( 例如, Wnddes 或边缘被插入或删除的图形) 。 每次更新操作后, 即持续输出解决方案的价值 。 我们通过持续观察( eval- 级别和用户级别) 对图表问题进行差异性私人算法, 即: 差异性私人动态图表算法 。 我们为部分动态计数问题提供事件一级私人算法, 如三角数计算, 使添加错误在艺术状态上得到改进( 美元更新序列的长度 美元) 。 在每次更新操作后首次算法中添加错误 $t. t. 我们还从 美元 的 数字级/ 水平, 将 美元 美元 的 水平 的 数字级 / 水平, 以 美元 美元 的 水平, 以 美元 美元 的 的 水平 和 最高 美元 的 美元 值 值 将 我们的 的 的 值 值 显示为 M 任何 。