Towards Privacy-Preserving Data Mining in Online Social Networks: Distance-Grained and Item-Grained Differential Privacy

Abstract

Online social networks have become increasingly popular, where users are more and more lured to reveal their private information. This brings about convenient personalized services but also incurs privacy concerns. To balance utility and privacy, many privacy-preserving mechanisms such as differential privacy have been proposed. However, most existent solutions set a single privacy protection level across the network, which does not well meet users’ personalized requirements. In this paper, we propose a fine-grained differential privacy mechanism for data mining in online social networks. Compared with traditional methods, our scheme provides query responses with respect to different privacy protection levels depending on where the query is from (i.e., is distance-grained), and also supports different protection levels for different data items (i.e., is item-grained). In addition, we take into consideration the collusion attack on differential privacy, and give a countermeasure in privacy-preserving data mining. We evaluate our scheme analytically, and conduct experiments on synthetic and real-world data to demonstrate its utility and privacy protection.

Appendix

Proof of Theorem 3: Assume that $V_1\sim Lap(\frac{1}{\epsilon _1})$, $V_2\sim Lap(\frac{1}{\epsilon _2})$, where $\epsilon _1 < \epsilon _2$. The conditional probability $\Pr [V_1|V_2]$ has a density function $\phi (x)$. Additionally, $V_1=h_{\epsilon _1}(x)=\epsilon _1\exp (-\epsilon _1|x|)$, $V_2=h_{\epsilon _2}(y)=\epsilon _2\exp (-\epsilon _2|y|)$. We use g(x, y) to denote the joint distribution density of $V_1$ and $V_2$. So g(x, y) holds:

$$\begin{aligned} g(x,y)=\phi (y-x)h(x) \end{aligned}$$

(3)

The density (3) should satisfy the following marginal distributions:

$$\begin{aligned} \int _{-\infty }^\infty g(x,y)dy=h_{\epsilon _1}(x)\quad \int _{-\infty }^\infty g(x,y)dx=h_{\epsilon _2}(y) \end{aligned}$$

(4)

The Eq. (4) could be seen as a convolution operation $\int _{-\infty }^\infty \phi (y-x)h_{\epsilon _1}(x)dx=h_{\epsilon _2}(y)$. We use Convolution Theorem to solve this equation:

$$\begin{aligned} \mathcal {F}_{\phi }(s)=\frac{\mathcal {F}_{h_{\epsilon _2}}(s)}{\mathcal {F}_{h_{\epsilon _1}}(s)}, \end{aligned}$$

(5)

where $\mathcal {F}$ denotes Fourier Transform. According to (5), we get:

$$\begin{aligned} \mathcal {F}_{\phi }(s)=\frac{\mathcal {F}_{h_{\epsilon _2}}(s)}{\mathcal {F}_{h_{\epsilon _1}}(s)}=\frac{1-\frac{s^2}{\epsilon _2^2}}{1-\frac{s^2}{\epsilon _1^2}} =(\frac{\epsilon _1}{\epsilon _2})^2(1+\frac{\epsilon _2^2-\epsilon _1^2}{\epsilon _1^2+s^2}) \end{aligned}$$

We set $b(x)=|x|^{1-\frac{n}{2}}K_{\frac{n}{2}-1}(|x|)$, where K denotes the modified Bessel function of the second kind, and $\mathcal {F}_{b}(s)=\frac{(2\pi )^\frac{n}{2}}{1+s^2}$. So,

$$\begin{aligned} \mathcal {F}^{-1}_{\phi }(s)=(\frac{\epsilon _1}{\epsilon _2})^2[\delta (x)+(\frac{\epsilon _2^2}{\epsilon _1^2}-1)\frac{\epsilon _1}{\sqrt{2\pi }}\sqrt{\epsilon _1 x}K_{-\frac{1}{2}}(\epsilon _1x)] \end{aligned}$$

$$\begin{aligned} \phi (x)\simeq (1-\frac{\epsilon _1^2}{\epsilon _2^2})\frac{\epsilon _1}{\sqrt{2\pi }}\sqrt{\epsilon _1 x}K_{-\frac{1}{2}}(\epsilon _1x) \end{aligned}$$

More relevant details about the proof are given by Koufogiannis et al. in [16].