# A Differentiated Anonymity Algorithm for Social Network Privacy Preservation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

**Definition**

**1.**

_{1}, A

_{2},…, A

_{n}} is a set of attribute nodes representing all users’ possible values of the privacy attribute, and A

_{i}corresponds to any privacy attribute values in a social network. For example, for the attribute called health-state, pneumonia and influenza, as two different property values form two different attribute nodes showed by A

_{1}= pneumonia, A

_{2}= influenza. S = {H, M, L}, is the set of the sensitivity of privacy attribute values; S(A

_{i}) represents the sensitivity of attribute value Ai, and Si is the short name of S(Ai), as shown in Figure 2.

**Definition**

**2.**

**Definition**

**3.**

_{1}, v

_{2}, …,v

_{n}} and d(v

_{i}) =|{u∈V : (u, v

_{i})∈E}|, and the type of attacker’s background knowledge F, the degree sequence of G is defined to be the sequence P = (d(v

_{1}), d(v

_{2}), …, d(v

_{n})). P can be divided into a group of subsequences [[d(1), …, d(i

_{1})], [d(i

_{1}+ 1), …, d(i

_{2})], …, [d(i

_{m}+ 1), …, P(j)]] such that G satisfies k-degree anonymity if, for every vertex v

_{i}∈V, there exist at least k–1 other vertices in G with the same degree as v

_{i}. In other words, for any subsequences P

_{y}= [d(i

_{y}+ 1), …, d(i

_{y+1})], P

_{y}satisfies two constraints: (1) All of the elements in P

_{y}share the same degree (d(i

_{y}+ 1) = d(i

_{y}+ 2) = … = d(i

_{y+1})); and (2) P

_{y}has size of at least k, namely (|i

_{y+1}− i

_{y}| ≥ k).

**Definition**

**4.**

_{i}’, there exists at least l–1 different attribute values in an equivalence group.

**Definition**

**5.**

## 3. The Differentiated Attribute-Preserving Model

#### 3.1. Graph Utility Measurement

#### 3.2. The Differentiated Attribute-Preserving Algorithm

#### 3.2.1. Basic Idea

#### Degree Centrality

#### Betweeness Centrality (BC)

_{st}is the number of the shortest path length between node s and node t and ${\mathrm{n}}_{\mathrm{st}}^{\mathrm{i}}$ is the number though node i. The index BC depicts the influence of nodes on the network information flow.

#### Closeness Centrality

_{ij}is the distance between node V

_{i}and node V

_{j}. In experiments, the parameter shows an effective description of a node’s topological importance.

#### Eigenvector Centrality

_{ij}) is the adjacency matrix. This shows not only a high number of neighbor nodes but is also an effective description of a node’s topological importance.

#### 3.2.2. Algorithm Framework

Algorithm 1. The DKDLD-UL Algorithm |

Input: Graph G(V, E, VA, S), k, l, F, and KV |

Output: DKDLD-UL anonymity graph G*. |

1. for(i = 1;i< = |V|;i++) |

2. { S.V_{i} = sensitivity(V_{i}); // Generate node attribute sensitivity |

3. if (S.V_{i} = M or H) then |

4. SV←V_{i}; // put V_{i} into privacy-preserving requirement vertices set SV |

5. end if } |

6. while (node V_{t} in SV) and (V_{t} is key node) do |

7. KV←V_{t}; // Generated attribute generalization node sequence set KV |

8. V_{t}.VA = [min, max] //min(max) is the minimal(maximum) attribute value in sequence Vt |

9. else if |

10. GV←V_{t}; // Generate node segmentation sequence set GV |

11. V_{t-1}, V_{t-2}←new node(V_{t}) // divide the current node into two new nodes |

12. For each social edge E_{i} of V_{t} |

13. Distribute by k-degree(E_{i}, V_{t-1}, V_{t-2}); // assign E_{i} to meet k-degree anonymity |

14. For each attribute VA_{i} of V_{t} |

15. Distribute by l-diversity(VA_{i}, V_{t-1}, V_{t-2}) // assign VA_{i} to meet l-diversity |

16. end while; |

17. Return G*; |

^{2}).

_{2},H), (3,2,A

_{3},H), (4,2,A

_{4},L), (1,1,A

_{1},M)] shown in Figure 3a, we get the DKDLD-UL anonymity graph G’ = [(2-1,2,A2,H), (3,2,A3,H), (4,2,A4,L), (1,1,A1,M), (2-2,1,A5,L)] shown in Figure 3b.

_{2}) = H, S(A

_{3}) = H, S(A

_{1}) = M,S(A

_{4}) = L, we obtain the privacy-preserving requirement node set SV = {V

_{2},V

_{3},V

_{1}}.

_{2}, V

_{3}, V

_{1}}.

_{2}in set GV = {V

_{2}, V

_{3}, V

_{1}}, the algorithm divides V

_{2}into child nodes V

_{2-1}, V

_{2-2}. For attribute values, it sets sensitivity attribute value for V

_{2-1}and sets non-sensitivity attribute value for node V

_{2-2}. For social edges, it divides the edges of node V

_{2}according to attribute similarity and remains the edge of other nodes, such as S(A

_{1}) = M, S(A

_{3}) = H, so edges (V

_{1},V

_{2}), (V

_{2},V

_{3}) are inherited by node V

_{2-1}, S(A

_{4}) = L, and edge (V

_{2},V

_{4}) is inherited by node V

_{2-2}. Therefore, we obtain edges = {(V

_{1}, V

_{2-1}), (V

_{2-1}, V

_{3}), (V

_{2-2}, V

_{4}), (V

_{3}, V

_{4})} in Figure 3b.

## 4. Experiments

#### 4.1. Datasets

#### 4.2. Results and Analysis

#### 4.2.1. Experiment 1: Running Time

#### 4.2.2. Experiment 2: Anonymity Cost

#### 4.2.3. Experiment 3: Utility Loss

## 5. Future Work and Limitations

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**An example of DKDLD-UL: An anonymous graph with node segmentation. (

**a**) The subgraph of node 2; (

**b**) the subgraph with node 2 atonomy.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Xie, Y.; Zheng, M.
A Differentiated Anonymity Algorithm for Social Network Privacy Preservation. *Algorithms* **2016**, *9*, 85.
https://doi.org/10.3390/a9040085

**AMA Style**

Xie Y, Zheng M.
A Differentiated Anonymity Algorithm for Social Network Privacy Preservation. *Algorithms*. 2016; 9(4):85.
https://doi.org/10.3390/a9040085

**Chicago/Turabian Style**

Xie, Yuqin, and Mingchun Zheng.
2016. "A Differentiated Anonymity Algorithm for Social Network Privacy Preservation" *Algorithms* 9, no. 4: 85.
https://doi.org/10.3390/a9040085