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：差分隐私算法通过确保用户数据的存在性与分析结果之间的弱相关性，在数据分析场景中保护个人隐私。动态图算法在持续演化的输入（即随时间插入或删除节点或边的图）上维持问题的解（如匹配），并在每次更新操作后持续输出解的值。本文研究在连续观测条件下（即差分隐私动态图算法）针对图问题的（事件级与用户级）差分隐私算法。针对三角形计数等部分动态计数型问题，我们提出的事件级隐私算法将现有技术的加性误差（以更新序列长度$T$为变量）降低了多项式因子，从而首次实现了加性误差为$T$的多对数阶。此外，我们针对最小生成树、最小割、最密子图和最大匹配问题，给出了$\varepsilon$-差分隐私的部分动态算法。其中改进的最小生成树算法的加性误差为$O(W \log^{3/2}T / \varepsilon)$（$W$为边权最大值），该下界被证明在$(\sqrt{\log T} / \varepsilon)$因子内紧致。针对其他问题，我们提出了一种部分动态算法，其乘法误差为$(1+\beta)$（$\beta>0$为任意常数），加性误差为$O(W \log(nW) \log(T) / (\varepsilon \beta) )$。最后，我们证明，对于一类具有用户级隐私的广义动态图算法，其加性误差必须与输出解取值范围的量级呈线性关系。