Local differential privacy (LDP for short), as an emerging standard privacy-preserving technique that is suitable for privacy preserving data analysis, has been widely deployed into various real-world scenarios to analyze massive data. In this study, we are mainly concerned with piecewise transformation technique (PTT for short) for analyzing numerical data under LDP. We first provide the principled framework for PTT in the context of LDP. Then, we show that (1) many PTTs are asymptotically optimal when used to obtain an unbiased estimator for mean of numerical data, and (2) there is PTT that reaches the theoretical lower bound on variance. Next, we prove that (1) there does not exist strictly better PTT than the well-used Duchi's scheme in terms of the consistency-noisy-variance, also, the latter is not always better than arbitrary PTT, (2) however, one has the ability to find a great number of PTTs that are consistently better than the latter according to the worst-case-noisy-variance. When we are restricted to the high privacy level$\epsilon \in (0,1]$, enough PTTs turn out to have smaller variance than the well-known Laplace mechanism. Lastly, we demonstrate that for a family of PTTs, the correspondingly theoretical lower bound of noisy variance follows$O(\epsilon ^{-2})$when considering$\epsilon \in (0,1]$.

In this work, we propose a principled framework using Vertex-based and Edge-based uniform generation mechanisms to build stochastic uniform growth tree networks that have a wide range of applications in various fields including physics, engineering, chemistry, ect., and then uncover the associated structural features analytically. When considering vertex-degree distribution, there exist three different classes of forms in the thermodynamic limit, i.e., exponential distribution, power-law distribution along with multiple-point distribution. At meantime, three distinct structural shapes are observed in the study of fractal phenomena, that is, fractal feature, critical phenomenon and non-fractal property. In addition, we obtain the analytical solution to fractal dimension for fractal structure from the probability point of view. More importantly, some well-known models, for instance, Vicsek fractal and T-graph, fall into our framework. Next, we precisely consider two families of stochastic uniform growth tree networks generated through the proposed framework. Specifically, we derive the analytic solution to mean hitting time$\langle \mathcal {H}\rangle$for measuring efficiency of delivering information on networks in a random-walk-based manner, and find that the introduction of randomness certainly enriches the scaling exponent of quantity$\langle \mathcal {H}\rangle$. Finally, we conduct extensive experiments, which suggests that computer simulations are in good agreement with theoretical analysis.