We propose an algorithm for counting below-threshold triangles in weighted graphs under local weight differential privacy. While prior work has largely focused on unweighted graphs, edge weights are intrinsic to many real-world networks. We consider the setting in which the graph topology is publicly known and privacy is required only for the contribution of an individual to incident edge weights, capturing practical scenarios such as road and telecommunication networks. Our method uses two rounds of communication. In the first round, each node releases privatized information about its incident edge weights under local weight differential privacy. In the second round, nodes locally count below-threshold triangles using this privatized information; we introduce both biased and unbiased variants of the estimator. We further develop two refinements: (i) a pre-computation step that reduces covariance and thus lowers expected error, and (ii) an efficient procedure for computing smooth sensitivity, which substantially reduces running time relative to a straightforward implementation. Finally, we present experimental results that quantify the trade-offs between the biased and unbiased variants and demonstrate the effectiveness of the proposed improvements.

翻译：我们提出了一种算法，用于在权重图中基于本地权重差分隐私对低于阈值的三角形进行计数。尽管先前的工作主要关注无权图，但边权重是许多真实网络的内在属性。我们考虑图拓扑结构公开、只需保护个体对邻接边权重的贡献隐私的设置，该设置可涵盖道路网络和电信网络等实际场景。我们的方法采用两轮通信。在第一轮中，每个节点在本地权重差分隐私下发布其邻接边权重的隐私化信息。在第二轮中，节点利用这些隐私化信息本地计算低于阈值的三角形；我们引入了有偏和无偏两种估计量变体。我们进一步开发了两项改进：（i）预处理步骤，用于减少协方差从而降低期望误差；（ii）一种高效计算平滑敏感度的过程，相较于直接实现方法可大幅减少运行时间。最后，我们通过实验结果量化了有偏与无偏变体间的权衡，并证明了所提改进的有效性。