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DS Decomposition Theory: Efficient Submodular Analysis in Social Networks

Explore the fundamental theorem and proofs of DS decompositions for submodular functions in social networks. Discover applications like viral marketing, rumor source detection, and composed influence studies. Learn the upper bounds and differences in influence spread. Delve into the concepts of active friending on LinkedIn and the Kempe-Kleinberg-Tardos Conjecture.

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DS Decomposition Theory: Efficient Submodular Analysis in Social Networks

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  1. DS Decomposition Lecture 2-1

  2. Fundamental Theorem MukundNarasimhan, Jeff A. Bilmes: A Submodular-supermodular Procedure with Applications to Discriminative Structure Learning. UAI 2005: 404-412 RishabhIyer and Jeff Bilmes: Algorithm for approximate Minimization of the difference between submodular functions, with application, arXiv:1207.0560v4 24 August 2013

  3. Fundamental Theorem

  4. Weak Version Proof

  5. Any Submodular Function Proof

  6. Fundamental Theorem Proof

  7. Open Problem • Given a set function, there are many DS decompositions for it. • Given a set function, can we find a DS function for it efficiently?

  8. Theorem Proof

  9. Proof

  10. Thank You, end

  11. Examples Lecture 1-2

  12. 1st Example Viral Marketing of Games

  13. Problem

  14. Examples

  15. Examples

  16. Upper Bound Lemma 1

  17. Lemma 2

  18. Wang, Zhefeng, et al. "Activity Maximization by Effective Information Diffusion in Social Networks." IEEE Transactions on Knowledge and Data Engineering 29.11 (2017): 2374-2387.

  19. 2nd Example Find Effector

  20. “Rumor” Source • Finding sources of activation is an important issue. • Given social network and diffusion model as well as all active nodes, find k effectors which best fit for the role of sources..

  21. Problem Formulation Theorem

  22. DS Decomposition

  23. References

  24. 3rd Example Composed Influence

  25. Problem • Two or more active persons together may give stronger influence than individual. • Composed influence can be formulated into hyper-edge. • With composed influence, the influence spread is neither submodular nor supermodular.

  26. Submodular Upper Bound • Create “super nodes” representing the start node of super-edges. • Add edges from nodes contained in super-node to the super nodes. • A super node is active if it contains at least one active node.

  27. Difference • What is the difference between upper bound and influence spread? • At least one super node is activated by an active node together with an inactive node.

  28. DS Decomposition of the difference • By the inclusive-exclusive formula, the difference between the upper bound and the influence spread can be expressed as MC IC

  29. Mutually-exclusive Cascade (MC)

  30. Mutually-exclusive Cascade

  31. Independent Cascade (IC)

  32. Independent Cascade

  33. Kempe-Kleinberg-Tardos Conjecture This conjecture is proved by Mossel and Roch in 2007 (STOC’07)

  34. The 4th Example Active Friending

  35. LinkedIn • Do you receive invitations from LinkedIn everyday? • Does LinkedIn have the following function: LinkedIn may suggest a list of invitations when you want to include a target person into friend list.

  36. Problem Formulation Theorem

  37. References

  38. Thank You, end

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