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：本文提出了一种在局部权重差分隐私下对加权图中阈值下三角形进行计数的算法。以往研究主要集中于无权图，而边权重是许多现实世界网络的固有属性。我们考虑图拓扑结构公开已知、仅需对个体对关联边权重的贡献进行隐私保护的场景，这涵盖了道路和通信网络等实际应用。我们的方法采用两轮通信：第一轮中，每个节点在局部权重差分隐私约束下发布其关联边权重的私有化信息；第二轮中，节点利用这些私有化信息在本地进行阈值下三角形计数，并提出了估计量的有偏和无偏两种变体。我们进一步开发了两项改进：（i）通过预计算步骤降低协方差从而减少期望误差；（ii）计算平滑敏感度的高效流程，相比直接实现显著降低了运行时间。最后，我们通过实验结果量化了有偏与无偏变体之间的权衡，并验证了所提改进措施的有效性。