1. TOWARDS IDENTITY ANONYMIZATION ON GRAPHS

2. INTRODUCTION

3. MOTIVATION Social networks, online communities, peer-to-peer file sharing and telecommunication systems can be modelled as complex graphs. These graphs are of significant importance in various application domains such as marketing, psychology and homeland security.

4. MOTIVATION in a social network, nodes correspond to individuals or other social entities, and edges correspond to social relationships between them. http://www.yasiv.com/facebook https://apps.facebook.com/touchgraph

5. THE CHALLENGE How to minimally modify the graph to protect the identity of each individual involved without losing the information ?

6. THE PRIVACY BREACHES IN SOCIAL NETWORK CATEGORIES identity disclosure: the identity of the individual who is associated with the node is revealed link disclosure: sensitive relationships between two individuals are disclosed content disclosure: the privacy of the data associated with each node is breached e.g., the email message sent and/or received by the individuals in a email communication graph

7. NOTES

8. PROBLEM

9. CHALLENGE we want to preserve the utility of the original graph, while at the same time satisfy the degree-anonymity constraint

10. PROBLEM DEFINITION

13. THE PROBLEM1

15. THE APPROACH

16. PROBLEM 2 FROM STEP 1 (DEGREE ANONYMIZATION)

17. PROBLEM 3 FROM STEP 2 (GRAPH CONSTRUCTION)

18. DEGREE ANONYMIZATION

20. For completeness, we also give a Greedy linear-time alternative algorithm for the Degree Anonymization problem. The Greedy algorithm first forms a group consisting of the first k highest-degree nodes. Then it checks whether it should merge the (k+1)th node into the previously formed group or start a new group at position (k + 1). For taking this decision the algorithm computes the following two costs:

21. GRAPH CONSTRUCTION

22. REALIZABILITY OF DEGREE SEQUENCE A degree sequence d, with d(1) ≥,..,..,.., ≥ d(n) is called realizable if and only if there exists a simple graph whose nodes have precisely this sequence of degrees. A degree sequence d with d (1) ≥ d (2) ≥… ≥ d (i) ≥… ≥ d (n) and Σ d (i) even, is realizable if and only if “Lemma 1”

23. THE CONSTRUCTGRAPH ALGORITHM: Takes as input the desired degree sequence d and outputs a graph with exactly this degree sequence, if such graph exists. Otherwise it outputs a “No" if such graph does not exist

24. REALIZABILITY OF DEGREE SEQUENCES WITH CONSTRAINTS

25. THE PROBING SCHEME If the Supergraph algorithm returns graph Ĝ, then we guarantee that the least number of edge additions has been made. If Supergraph returns “No” or “Unknown”, we are content in tolerating some more edge-additions in order to get the Probing scheme that forces the Supergraph algorithm to output the desired k-degree anonymous graph with a little extra cost.

26. THE PROBING SCHEME

27. RELAXED GRAPH CONSTRUCTION Most of the edges of the original graph appear in the degree- anonymous graph as well, but not necessarily all of them.

28. RELAXED GRAPH CONSTRUCTION

29. The Greedy_Swap algorithm

30. RELAXED GRAPH CONSTRUCTION The Priority algorithm a simple modification of the ConstructGraph algorithm that allows the construction of degree anonymous graphs with similar high edge intersection with the original graph directly, without using Greedy_Swap it gives priority to already existing edges in the input graph G(V;E).

31. RELAXED GRAPH CONSTRUCTION

32. EXPERIMENTS

33. EVALUATING DEGREE ANONYMIZATION ALGORITHMS The closer R is to 1, the better the performance of the Greedy algorithm

34. EVALUATING GRAPH CONSTRUCTION ALGORITHMS Evaluating Anonymization cost L1(dA - d) The smaller the value of L1(dA - d) the better the qualitative performance of the algorithm.

35. EVALUATING GRAPH CONSTRUCTION ALGORITHMS Clustering Coefficient (CC): We additionally compare the clustering coefficients of the anonymized graphs with the clustering coefficients of the original graphs.

36. EVALUATING GRAPH CONSTRUCTION ALGORITHMS Average Path Length (APL):