With the growth of online social services, social information graphs are becoming increasingly complex. Privacy issues related to analyzing or publishing on social graphs are also becoming increasingly serious. Since the shortest paths play an important role in graphs, privately publishing the shortest paths or distances has attracted the attention of researchers. Differential privacy (DP) is an excellent standard for preserving privacy. However, existing works to answer the distance query with the guarantee of DP were almost based on the weight private graph assumption, not on the paths themselves. In this paper, we consider edges as privacy and propose distance publishing mechanisms based on edge DP. To address the issue of utility damage caused by large global sensitivities, we revisit studies related to asymmetric neighborhoods in DP with the observation that the distance query is monotonic in asymmetric neighborhoods. We formally give the definition of asymmetric neighborhoods and propose Individual Asymmetric Differential Privacy with higher privacy guarantees in combination with smooth sensitivity. Then, we introduce two methods to efficiently compute the smooth sensitivity of distance queries in asymmetric neighborhoods. Finally, we validate our scheme using both real-world and synthetic datasets, which can reduce the error to $0.0862$.

翻译：随着在线社交服务的增长，社交信息图正变得越来越复杂。与社交图分析或发布相关的隐私问题也日益严重。由于最短路径在图结构中扮演重要角色，以隐私保护方式发布最短路径或距离已引起研究者的关注。差分隐私（DP）是一种优秀的隐私保护标准。然而，现有在DP保证下回答距离查询的研究几乎都基于权重隐私图假设，而非路径本身。本文以边作为隐私保护对象，提出基于边差分隐私的距离发布机制。针对全局敏感度过大导致的效用损失问题，我们重新审视了DP中与不对称邻域相关的研究，并观察到距离查询在不对称邻域中具有单调性。我们正式给出了不对称邻域的定义，并结合平滑敏感度提出了具有更高隐私保证的个体不对称差分隐私。随后，我们引入两种方法高效计算不对称邻域中距离查询的平滑敏感度。最后，我们通过真实数据集和合成数据集验证了所提方案，可将误差降低至$0.0862$。