In this paper, we give an algorithm to publish the number of paths and Katz centrality under the local differential privacy (LDP), providing a thorough theoretical analysis. Although various works have already introduced subgraph counting algorithms under LDP, they have primarily concentrated on subgraphs of up to five nodes. The challenge in extending this to larger subgraphs is the cumulative and exponential growth of noise as the subgraph size increases in any publication under LDP. We address this issue by proposing an algorithm to publish the number of paths that start at every node in the graph, leading to an algorithm that publishes the Katz centrality of all nodes. This algorithm employs multiple rounds of communication and the clipping technique. Both our theoretical and experimental assessments indicate that our algorithm exhibits acceptable bias and variance, considerably less than an algorithm that bypasses clipping. Furthermore, our Katz centrality estimation is able to recall up to 90% of the nodes with the highest Katz centrality.

翻译：本文提出了一种在本地差分隐私（LDP）框架下发布路径数量与Katz中心性的算法，并提供了详尽的理论分析。尽管已有多种工作提出了基于LDP的子图计数算法，但这些研究主要集中于节点数不超过五个的子图。将该方法扩展到更大规模子图的难点在于：在LDP机制下，任何发布过程中随着子图规模增大，噪声会呈现累积且指数级增长。针对这一问题，我们通过提出一种发布图中每个节点出发路径数量的算法，进而实现所有节点Katz中心性的发布。该算法采用多轮通信机制与剪裁技术。理论分析与实验评估均表明：与未采用剪裁技术的算法相比，本算法的偏差与方差均处于可接受水平且显著更低。此外，我们的Katz中心性估计能够正确召回高达90%的具有最高Katz中心性的节点。