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Lecture 2-5 Kempe-Kleinberg-Tardos Conjecture A simple proof

Lecture 2-5 Kempe-Kleinberg-Tardos Conjecture A simple proof. Ding-Zhu Du University of Texas at Dallas. Outline of KKT Conjecture. IM in Threshold Model KKT Conjecture A Simple Proof Applications. General Threshold Model. 1. 2. 3. LT:. Inactive Node. Y. 0.6. Active Node.

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Lecture 2-5 Kempe-Kleinberg-Tardos Conjecture A simple proof

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  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

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