Understanding Kempe-Kleinberg-Tardos Conjecture: A Simple Proof and Applications

1. Lecture 2-5 Kempe-Kleinberg-Tardos Conjecture A simple proof Ding-Zhu Du University of Texas at Dallas

2. Outline of KKT Conjecture • IM in Threshold Model • KKT Conjecture • A Simple Proof • Applications

3. General Threshold Model 1 2 3

4. LT: Inactive Node Y 0.6 Active Node Threshold 0.2 0.2 0.3 Active neighbors X 0.1 0.4 U 0.3 0.5 Stop! 0.2 0.5 w v

5. Inapproximability Theorem Proof Recall

6. Input size

7. is monotone submodular. is monotone but not submodular.

8. Outline • Threshold Model • KKT Conjecture • A Simple Proof • Applications

9. Kempe-Kleinberg-Tardos Conjecture This conjecture is proved by Mossel and Roch in 2007 (STOC’07)

10. Linear Threshold (LT) Model • A node v has random threshold ~ U[0,1] • A node v is influenced by each neighbor w according to a weight bw,v such that • A node v becomes active when at least (weighted) fraction of its neighbors are active

11. Example Inactive Node Y 0.6 Active Node Threshold 0.2 0.2 0.3 Active neighbors X 0.1 0.4 U 0.3 0.5 Stop! 0.2 0.5 w v

12. Outline • Threshold Model • KKT Conjecture • A Simple Proof • Applications

13. Idea: Piecemeal Growth Seeds can be distributed step by step or altogether, the distribution of final influence set does not change.

14. Notations ~

15. Notations ~ ~ ~ ~

16. Lemma 1 Proof

17. 1st Try ~ ~

18. 1st Try

19. 1st Try Not true!

20. More Techniques:Antisense Phase and Need-to-Know Representation

21. Antisense Phase ~ ~

22. Lemma 2 ~ ~ Proof

23. 2nd Try ~ ~

24. 2nd Try

25. Lemma 3 Proof

26. Outline • Threshold Model • KKT Conjecture • A Simple Proof • Applications

27. Linear Threshold (LT) Model • A node v has random threshold ~ U[0,1] • A node v is influenced by each neighbor w according to a weight bw,v such that • A node v becomes active when at least (weighted) fraction of its neighbors are active

28. Decreasing Cascade Model

29. Deterministic Model

30. Independent Cascade

31. Triggering Model

32. IC is a special cases of Triggering Model

33. LT(or MC) is a special cases of Triggering Model

34. Triggering Model • Triggering model is not a general threshold model. • When triggering set at every node is fixed, it can be seen as a threshold model. Then # of influenced nodes is a monotone increasing submodular function of seed set. • Triggering model is a linear combination of threshold models. Coefficients are probability.

35. “Only-Listen-Once” Model

36. References 1 2

37. 1 2 3

38. 1 1 2 3 2 3 1 1 2 3 2 3

39. 1 2 3

40. Proof of Submodularity

41. Conjecture

42. THANK YOU!

43. Open Problem 1