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Lecture 2-2 Independent Cascade Model and Linear Threshold Model

Lecture 2-2 Independent Cascade Model and Linear Threshold Model. Ding-Zhu Du University of Texas at Dallas. Independent Cascade (IC) Model. When node v becomes active, it has a single chance of activating each currently inactive neighbor w.

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Lecture 2-2 Independent Cascade Model and Linear Threshold Model

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  1. Lecture 2-2 Independent Cascade Model and Linear Threshold Model • Ding-Zhu Du • University of Texas at Dallas

  2. Independent Cascade (IC) Model • When node v becomes active, it has a single chance of activating each currently inactive neighbor w. • The activation attempt succeeds with probability pvw . • The deterministic model is a special case of IC model. In this case, pvw =1 for all (v,w).

  3. Example Y 0.6 Inactive Node 0.2 0.2 0.3 Active Node Newly active node U X 0.1 0.4 Successful attempt 0.5 0.3 0.2 Unsuccessful attempt 0.5 w v Stop!

  4. Independent Cascade (IC)

  5. 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

  6. 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

  7. A property

  8. Mutually-exclusive Cascade (MC)

  9. Theorem Proof

  10. Acyclic Case

  11. General Case In linear threshold model

  12. General Case In Mutually-exclusive Cascade model

  13. Mutually-exclusive Cascade (MC)

  14. Examples

  15. 1st Example difference between IC and LT

  16. 1 2 3

  17. 1 1 2 3 2 3 1 1 2 3 2 3

  18. 1 2 3

  19. 1 2 3

  20. 1 2 3

  21. 2nd Example A property of LT=MC

  22. 1 2 3 1 1 2 3

  23. Equivalent Networks

  24. Influence Maximization Problem • Influence spread of node set S: σ(S) • expected number of active nodes at the end of diffusion process, if set S is the initial active set. • Problem Definition (by Kempe et al., 2003): (Influence Maximization). Given a directed and edge-weighted social graph G = (V,E, p), a diffusion model m, and an integer k ≤ |V |,find a set S ⊆ V , |S| = k, such that the expected influence spread σm(S) is maximum.

  25. Known Results • Bad news: NP-hard optimization problem for both IC and LT models. • Good news: • σm(S) is monotone and submodular. • We can use Greedy algorithm! • Theorem: The resulting set S activates at least (1-1/e) (>63%) of the expected number of nodes that any size-k set could activate .

  26. Proof of Submodularity

  27. Decision Version of InfMax in IC Is it in NP? Theorem Corollary

  28. Disadvantage • Lack of efficiency. • Computing σm(S) is # P-hard under both IC and LT models. • Selecting a new vertex u that provides the largest marginal gain σm(S+u) - σm(S), which can only be approximated by Monte-Carlo simulations (10,000 trials). • Assume a weighted social graph as input. • How to learn influence probabilities from history?

  29. Monte-Carlo Method Buffon's needle

  30. References

  31. THANK YOU!

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