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

Explore a straightforward proof of the Kempe-Kleinberg-Tardos Conjecture within the context of the Linear Threshold Model. Learn about the inactive node concept, influence thresholds, and applications of the theory in various models. The text outlines the necessary proofs and applications, providing insights into the behavior of different models and their implications.

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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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