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On the target set selection problem

On the target set selection problem. 江俊瑩. Potential Customers. Free Samples. Word-of-mouth. Marketing. Marketing. Contagion. A Social Network with Threshold Function. 2. 1. 3. 4. 3. 1. 1. 2. 2. 2. Target Set S. 2. 1. 3. 4. 3. 1. 1. 2. 2. 2.

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On the target set selection problem

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  1. On the target set selection problem 江俊瑩

  2. Potential Customers

  3. Free Samples

  4. Word-of-mouth

  5. Marketing

  6. Marketing

  7. Contagion

  8. A Social Network with Threshold Function 2 1 3 4 3 1 1 2 2 2

  9. Target Set S 2 1 3 4 3 1 1 2 2 2

  10. Activation Process Starting form S 2 1 3 4 3 1 1 2 2 2

  11. min-seed(G,θ) 2 1 3 4 3 1 1 2 2 2

  12. TARGET SET SELECTION TARGETSETSELECTION:

  13. Threshold Models • Constant threshold: for all vertices v in G. • Majority threshold : for all vertices v in G. • Strict majority threshold : for all vertices v in G.

  14. Parallel updating rule: All white vertices v that have at least black neighbors at the previous round are colored black. The colors of the other vertices do not change. Sequential updating rule: Exactly one of white vertices that have at least black neighbors at the previous round is colored black. The colors of the other vertices do not change. Updating rules

  15. Lemma: Let be a connected graph G with thresholds on V(G). An optimal target set for under the sequential updating rule is also an optimal target set for under the parallel updating rule, and vice versa. Parallel = Sequential

  16. A Lower bound for min-seed(G,θ)

  17. [Dreyer and Roberts, 2009] In constant threshold model, it is NP-hard to compute the min-seed for any . Bad News [Peleg, 2002] It is NP-hard to compute the optimal target set for majority thresholds. [Chen Ning, 2009] The TARGET SET SELECTION problem is NP-hard when the thresholds are at most 2.

  18. [Chen Ning, 2009] Given any regular graph with thresholds for any vertex v, the TARGET SET SELECTION problem can not be approximatedwithin the ratio of , for any fixed constant , unless Extremely Bad News !!

  19. [Ben-Zwi et al, 2010] For n-vertices graph G with treewidth bounded by , the TARGET SET SELECTION problem can be solved in time. Results for Trees [Dreyer and Roberts, 2009] When G is a tree, the TARGET SET SELECTION problem can be solved in linear time for constant thresholds. [Chen Ning, 2009] When the underlying graph is a tree, the problem can be solved in polynomial-time under a general threshold model.

  20. Block-cactus graph

  21. v v vertex-sum at v of G1and G2 G1 ⊕v G2 v G1 G2

  22. v (G1 ⊕v G2 , θ)

  23. v

  24. 1 2 3 2 2 2 5 2 5 3 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set

  25. 1 2 3 2 2 2 5 2 5 3 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set

  26. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set

  27. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  28. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  29. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  30. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  31. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  32. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  33. 2 b c 4 4 a 1 2 3 2 2 2 2 5 d 2 5 3 e 5 1 1 3 5 2 2 2 2 2 6 2 2 2 2 2 1 1 Optimal Target Set 2 b 2 5 d e 4 c a 4

  34. 2 b c 4 4 a 2 d e 5 Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 5 2 2 2 2 2 2 6 b 2 5 2 d 2 e 2 2 2 4 c a 4 1 1

  35. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 5-2 2 2 2 2 2 6 2 2 2 2 2 1 1

  36. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 2 2 2 2 2 6 2 2 2 2 2 1 1

  37. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 2 2 2 2 2 6 2 2 2 2 2 1 1

  38. Optimal Target Set 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 2 5 2 5 3 1 1 3 3 2 2 2 2 2 6 2 2 2 2 2 1 1

  39. Optimal Target Set S1 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 2 5 2 5 3 1 1 3 3 2 2 2 2 2 6 2 2 2 2 2 1 1

  40. Optimal Target Set S1 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 2 5 3 1 1 3 3 2 2 2 2 2 6 2 2 2 2 2 1 1

  41. Optimal Target Set S1 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 2 5 0 1 2 2 1 2 1201 2 2 1 3 1 1 3 3 2 2 2 2 2 6 2 2 2 2 2 1 1

  42. Optimal Target Set S1 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 2 5 0 1 2 2 1 2 1201 2 2 1 3 1 1 3 3 2 2 2 2 2 6 1 2 2 1 2 11 2 2 1 2 2 2 2 2 1 1

  43. Optimal Target Set S1 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 2 5 0 1 2 2 1 2 1201 2 2 1 3 1 1 3 3 2 2 2 2 2 6 1 2 2 1 2 11 2 2 1 2 2 2 2 2 1 1

  44. Optimal Target Set S1 1 2 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 2 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 2 5 0 1 2 2 1 2 1201 2 2 1 3 1 1 3 3 2 2 2 2 2 6 1 2 2 1 2 11 2 2 1 2 2 2 2 2 1 1

  45. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 2-1 2 2 2 2 6 2 2 2 2 2 1 1

  46. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 1 2 2 2 2 6 2 2 2 2 2 1 1

  47. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 1 2 2 2 2 6 2 2 2 2 2 1 1

  48. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 1 2 2 2 2 6 2 2 2 2 2 1 1

  49. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 1 2 2 2 2 6 2 2 2 2 2 1 1

  50. Optimal Target Set 1 2 3 2 2 2 5 2 5 3 1 1 3 3 1 2 2 2 2 6 2 2-0 2 2 2 1 1

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