1. Private Release of Graph Statistics using Ladder Functions J.ZHANG, G.CORMODE, M.PROCOPIUC, D.SRIVASTAVA, X.XIAO

2. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

3. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

4. Data Release companyinstitute public adversary

5. Private Data Release private algorithm user Objective 1: the noisy answer should reveal little about any individual in the database Objective 2: the noisy answer should be as accurate as possible

6. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

7. Differential Privacy on Graph Differentially private algorithm injects noise into the query answer, in order to cover the maximum impact of a relationship (an edge). Differentially private answer: binary answer + 1*noise 1 0

8. Differential Privacy on Graph Differentially private algorithm injects noise into the query answer, in order to cover the maximum impact of a relationship (an edge). Query: how many edges? Differentially private answer: true answer + 1*noise 6 5 neighboring 1514

9. Differential Privacy on Graph Differentially private algorithm injects noise into the query answer, in order to cover the maximum impact of a relationship (an edge). Query: how many triangles? Differentially private answer: true answer + 4*noise 2016 neighboring If there are n nodes: true answer + (n-2)*noise

10. Differential Privacy on Graph Differentially private algorithm injects noise into the query answer, in order to cover the maximum impact of a relationship (an edge). 4 In a real dataset CondMat from SNAP, #triangle = 173,361 while n-2 = 23,131 Query: how many triangles? Differentially private answer: true answer + 4*noise

11. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

12. Formal Definition [TCC’06]

14. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

15. Global Sensitivity [TCC’06]

16. Global Sensitivity

19. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

20. Local Sensitivity [STOC’07] GS = 4 LS(g) = 1 OR

21. Global Sensitivity

23. Local Sensitivity

26. GS = 4 LS(g) = 1 OR

27. Local Sensitivity

28. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

29. Ladder Functions We change slope gradually.

30. Ladder Functions We change slope gradually.

31. Ladder Functions We change slope gradually.

32. Ladder Functions: Summary ladder functions

33. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

34. Formal Results *In the paper, we present a discrete version of ladder functions

35. Graph Statistics triangle counting

36. Graph Statistics triangle counting

37. Formal Results For efficient sampling, we adopt Exponential Mechanism (pls refer to Section 3.2 of the paper).

38. Our Contributions triangle counting These two queries have been solved before. The solutions are also local sensitivity based. But they are either achieving a weakened version of differential privacy or less accurate. Our solution is the most accurate, with a pure differential privacy guarantee.

39. Our Contributions triangle counting Our solution is the first local sensitivity based solution with pure differential privacy guarantee. Our solution is the most accurate.

40. Overview  The Problem: Private Release of Graph Statistics  Differential Privacy on Graph  Two “Solutions”: Global Sensitivity (GS) and Local Sensitivity (LS)  Global Sensitivity  Local Sensitivity  Ladder Functions: From LS to GS  Formal Results and Contributions  Experiments

41. Experiment Results: CondMat triangle counting

42. Conclusions  We show that the pure differential privacy guarantee can be obtained for graph statistics, specifically subgraph counts.  We propose ladder functions which combine the merits of both GS and LS.  Future work includes extending the ladder framework to  large scale graphs like Facebook friendship graph  functions outside the domain of graphs, e.g., median of an array, machine learning tasks Thank you!