# Sensitivity Estimation for Differentially Private Query Processing
## 5 Experiments
### Impact of Parameters  
Impact of Sample Size. The key factor of our Sampling-SE method is the sample rate. We conducted experiments to see how the sample rate \(r\) affects the noise results. For simplicity, we took two datasets and two chain queries (Q1 and Q4) for this experiment. For the parameters, we used the following: scale = 1 GB; \(\epsilon=6.4\); and \(\delta=2\times 10^{-10}\) for TPC-H, whereas we used \(\delta=10^{-7}\) for the Facebook dataset. Moreover, different noise mechanisms, including the _Laplace mechanism_[8] and _General Cauchy mechanism_[9], were implemented to confirm our results. As shown in Figure 10, Sampling-SE significantly outperforms ES in accuracy across different differential privacy mechanisms. As sampling size increases, the noise level of Sampling-SE converges closer to the ground truth results. Furthermore, the Laplace mechanism consistently yields lower noise levels than the General Cauchy mechanism for all methods, including both baseline approaches and our proposed technique.  
Figure 9: Running time of queries on the Facebook dataset.  
To further evaluate the efficacy of our sampling method, we examined a cyclic residual query, representing a key target scenario for Sampling-SE: \"select max(count) from (select edge2_from, count(*) from \(edge2\)\(\times\)\(edge3\)\(\times\)\(edge4\)\(\times\)\(edge5\) group by edge2_from) as t,\". Figure 11 shows that even at low sampling rates, varying from \(10^{-5}\) to \(10^{-4}\), the method achieves highly accurate estimations. This confirms the effectiveness of Sampling-SE for complex cyclic queries while maintaining computational efficiency.  
Figure 11: Estimate results under different sampling rates.  
Figure 10: Noise level under different sample rates.  
Additionally, we tested the effect of sampling rate on sensitivity as a way to eliminate the interference of the differential privacy noise added to the data. The experimental result is shown in Figure 12. To enhance efficiency for high-cardinality join results, we introduced Sampling-SE-WithFilter, a strategy that estimates only the maximum group size via random sampling rather than computing all group sizes. As illustrated, this approach achieves significantly faster convergence than basic Sampling-SE with increasing sampling rates.  
Impact of Data scale. In this part, we evaluated the effects of data scale on the noise level. We used the TPC-H datasets with scale factors ranging from 0.01 to 10 and tested the performance of different methods with Q1, Q2, and Q3. We can see from Figure 13 that Sampling-SE maintains stable noise levels as data scales, confirming its suitability for large datasets. Note that for Q1 and Q2, Sampling-SE preserves the maximum boundaries of RS, yielding identical noise levels. To validate utility preservation, we compared noise magnitudes with the original query results. Crucially, Sampling-SE maintains noise levels strictly below the actual query answers, guaranteeing the original data insights remain clearly visible, which enables practical deployment in real-world applications where data utility is mission-critical.  
Figure 12: Sensitivity under different sampling rates.  
Figure 13: Impact of data scale on the noise level.