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Understanding Independent Cascade and Linear Threshold Models

Explore the Independent Cascade (IC) and Linear Threshold (LT) models in social graph dynamics, influence maximization, and implications for diffusion processes. Learn about activation probabilities, threshold influences, and the application of these models in network analysis.

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Understanding Independent Cascade and Linear Threshold Models

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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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