A Maximum Variance Approach for Graph Anonymization

Abstract

Uncertain graphs, a form of uncertain data, have recently attracted a lot of attention as they can represent inherent uncertainty in collected data. The uncertain graphs pose challenges to conventional data processing techniques and open new research directions. Going in the reserve direction, this paper focuses on the problem of anonymizing a deterministic graph by converting it into an uncertain form. The paper first analyzes drawbacks in a recent uncertainty-based anonymization scheme and then proposes Maximum Variance, a novel approach that provides better tradeoff between privacy and utility. Towards a fair comparison between the anonymization schemes on graphs, the second contribution of this paper is to describe a quantifying framework for graph anonymization by assessing privacy and utility scores of typical schemes in a unified space. The extensive experiments show the effectiveness and efficiency of Maximum Variance on three large real graphs.

A Proof of Theorems

1.1 A.1 Proof of Theorem 1

Proof

We prove the result by induction.

When $k=1$, we have two cases of $G_1$: $E_{G_1}=\{e_1\}$ and $E_{G_1}=\emptyset$. For both cases, $Var[D(\mathcal {G}_1,G_1)] = p_1(1-p_1)$, i.e. independent of $G_1$.

Assume that the result is correct up to $k-1$ edges, i.e. $Var[D(\mathcal {G}_{k-1},G_{k-1})] = \sum _{i=1}^{k-1} p_i(1-p_i)$ for all $G_{k-1} \sqsubseteq \mathcal {G}_{k-1}$, we need to prove that it is also correct for $k$ edges. We use the subscript notations $\mathcal {G}_k, G_k$ for the case of $k$ edges. We consider two cases of $G_k$: $e_k \in G_k$ and $e_k \notin G_k$.

Case 1. The formula for $Var[D(\mathcal {G}_k, G_k)]$ is

$$\begin{aligned} Var[D(\mathcal {G}_k,G_k)] = \sum _{G'_k \sqsubseteq \mathcal {G}_k} Pr(G'_k) [D(G'_k,G_k) - E[D(\mathcal {G}_k,G_k)]]^2 \nonumber \\ = \sum _{e_k \in G'_k} Pr(G'_k) [D(G'_k,G_k) - E[D_k]]^2 + \sum _{e_k \notin G'_k} Pr(G'_k) [D(G'_k,G_k) - E[D_k]]^2 \nonumber \end{aligned}$$

The first sum is $\sum _{G'_{k-1} \sqsubseteq \mathcal {G}_{k-1}} p_k Pr(G'_{k-1})[D_{k-1} - E[D_{k-1}] - (1-p_k)]^2$.

The second sum is $\sum _{G'_{k-1} \sqsubseteq \mathcal {G}_{k-1}} (1-p_k) Pr(G'_{k-1})[D_{k-1} - E[D_{k-1}] + p_k)]^2$.

Here we use shortened notations $D_k$ for $D(G'_k,G_k)$ and $E[D_k]$ for $E[D(\mathcal {G}_k,G_k)]$.

By simple algebra, we have $Var[D(\mathcal {G}_k,G_k)] = Var[D(\mathcal {G}_{k-1},G_{k-1})] + q_k(1-q_k) = \sum _{i=1}^{k} p_i(1-p_i)$.

Case 2. similar to the Case 1.    $\square$