# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
[20] proposed a differential privacy frequent itemsets mining algorithm and had precisely quantified the trade-off between privacy and utility in frequent itemsets mining. Search log using differential privacy can get the trade-off. [PERSON] et al. [21] proposed a sanitization framework to perfectly tackle the trade-off in the context of search logs in a distributed manner. Differential privacy can guarantee the trade-off in statistics analysis. Within Bayesian inference based on differential privacy under certain conditions on the prior, the sampling from the posterior distribution could lead to a desired level of privacy and utility [22]. In data aggregation, keeping trade-off using differential privacy is very important. [PERSON] and [PERSON] [23] proposed two methods: single to randomized multiple dummies, and single to randomized multiple dummies with Bayes, both of which generated a set of disguised values of one sensed value of anonymization and ensured a better trade-off between privacy and utility. Differential privacy can maintain the trade-off in data query. [PERSON] et al. [24] studied the trade-off in the context of counting queries and more general linear queries. Data releasing using differential privacy has achieved the trade-off. [PERSON] et al. [25] proposed two novel mechanisms achieving the trade-off between the utility of the query results and the degree of privacy preservation, namely NoiseFirst determining the histogram structure after the noise injection and StructureFirst of deriving the histogram structure before the addition of the noise. Differential privacy can achieve the trade-off in location-based service and recommendation service. Local differential perturbation mechanisms could provide a well-balanced trade-off between location privacy and service accuracy [26]. [PERSON] et al. [27] proposed a mechanism to achieve differential privacy graph-link analysis-based social recommendation. Differential privacy can also guarantee the trade-off in other applications fields. [PERSON] et al. [28] adopted differential privacy aggregates of web browsing activities that could be realized in real time while preserving the utility of shared data. [PERSON] et al. [29] proposed an efficient distortion-based privacy-preserving metering scheme that protected an individual customer's privacy and provided the complete power consumption distribution curve of a multitude of customers without privacy invasion. [PERSON] et al. [30] proposed differential privacy crowdsourced spectrum sensing that allowed the spectrum-sensing provider to select mobile users for executing spatiotemporal spectrum-sensing tasks without violating the location privacy of mobile users. [PERSON] and [PERSON] [31] showed that the exponential mechanism could be implemented as a truthful mechanism while still preserving differential privacy for any mechanism design problem with the objective of maximizing social welfare. However, there is no unified trade-off measurement of differential privacy mechanisms and their applications. This problem is covered in the definition of privacy-preserving monotonicity. According to the privacy metrics and utility metrics of this paper, the trade-off of differential privacy means that privacy (utility) increases and utility (privacy) decreases. Thus, the increasing of privacy-preserving level leads to the decreasing of data utility as privacy budget decreases in differential privacy mechanisms. Conversely, the decreasing of privacy-preserving level leads to the increasing of data utility as privacy budget increases. Current works have no unified metrics of measuring the trade-off of differential privacy. It is imperative to seek a unified measurement as an evaluation criterion of the trade-off between privacy and utility in differential privacy. Thus, we proposed the definition of privacy-preserving monotonicity of differential privacy to formulate the trade-off between privacy and utility based on privacy metrics and utility metrics of this paper.