We propose an algorithm for counting below-threshold triangles in weighted graphs under local weight differential privacy. While prior work focused on unweighted graphs, many real-world networks naturally include edge weights. We study the setting where the graph topology is public known and the privacy of the influence of an individual on the edge weights is protected. This captures realistic scenarios such as road networks and telecommunication networks. Our approach consists of two rounds of communication. In the first round, each node publishes their incident weight information under local weight differential privacy while in the second round, the nodes locally count below-threshold triangles, for which we introduce a biased and unbiased variant. We further propose two different improvements. We present a pre-computation step that reduces the covariance and thereby lowers the expected error. Secondly, we develop an algorithm for computing the smooth-sensitivity, which significantly reduces the running time compared to a straightforward approach. Finally, we provide experimental results that demonstrate the differences between the biased and unbiased variants and the effectiveness of the proposed improvements.

翻译：我们提出了一种在局部权重差分隐私下对加权图中低于阈值的三角形进行计数的算法。先前的研究主要集中于无权图，而许多现实世界的网络天然包含边权重。我们研究图拓扑结构公开已知，且个体对边权重影响的隐私受到保护的场景。这捕捉了道路网络和电信网络等现实场景。我们的方法包含两轮通信。在第一轮中，每个节点在局部权重差分隐私下发布其关联边的权重信息；在第二轮中，节点本地计数低于阈值的三角形，为此我们引入了有偏和无偏两种变体。我们进一步提出了两种不同的改进方案。我们提出了一个预计算步骤，该步骤降低了协方差，从而减少了期望误差。其次，我们开发了一种计算平滑敏感度的算法，与直接方法相比，该算法显著降低了运行时间。最后，我们提供了实验结果，展示了有偏和无偏变体之间的差异以及所提改进方案的有效性。