lidong.wu@utdallas

1. Lecture 5-2 Target-Influence and Power Law Graphs Ding-Zhu Du Univ of Texas at Dallas lidong.wu@utdallas.edu

2. Product Advertisement

3. More Companies: Competition

4. How to Measure Influence?

5. A Winning Strategy:Get enough Positive Influence Majority!!!

6. Positive-Influence Max x B A

7. Minimum Budget Maximum Influence • Given • a market (e.g. a set of individuals) • estimates for influence between individuals • Goal • Minimum budget for initial advertising (e.g. give away free samples of product) in order to occupy the market. • Question • Which set of individuals should we target at? • Application besides product marketing • spread an innovation, ideas, news • detect stories in blogs • analyze Twitter

8. Positive-Influence Dominating:Min Budget • Given a network, • Find a minimum positive-influence dominating set.

9. Some Product may target only a portion of networks!

10. Target-Dominating • Given a network G=(V,E) and a node subset Q, • Find a minimum node-subset positive-influence dominating Q. • Q is called a target set. • How can a boy to influence a girl? (He Chen et al.)

11. Potential Function Lemma 1 Proof 1 2 3

12. Greedy Algorithm

13. Theorem

14. Lemma 1 2

15. Theorem

16. Theorem

17. Power Law Graph During the evolution and growth of a network, the great majority of new edges are to nodes with an already high degree.

18. Power-law distribution Log-log scale: log f(x) ~ –αlog x Power law distribution: f(x) ~ x–α

19. Power Law • Nodes with high degrees may have “butterfly effect”. • Small number • Big influence

22. Theorem

23. Lemma 1 Lemma 2 Proof

24. Lemma 3 Proof

25. Theorem Proof

26. Remarks on Power-Law Networks • Constant-approximation is easily obtained. • NP-hardness is not so hard to prove. • Usually, it is open for existence of PTAS.

27. Remarks on Power-Law Networks • Could we really ignore small difference? 你懂的

28. Open Problems • Is there a constant-approximation for any target set in power-law graphs? • Is there a constant-approximation for positive-influence max in power-law graphs?

29. References • Feng Zou, Zhao Zhang, Weili Wu: Latency-Bounded Minimum Influential Node Selection in Social Networks. WASA 2009: 519-526 • Feng Zou, James Willson, Zhao Zhang, Weili Wu: Fast Information Propagation in Social Networks. Discrete Math., Alg. and Appl. 2(1): 125-142 (2010)

30. Feng Wang, Hongwei Du, Erika Camacho, Kuai Xu, Wonjun Lee, Yan Shi, Shan Shan: On positive influence dominating sets in social networks. Theor. Comput. Sci. 412(3): 265-269 (2011) • Thang N. Dinh, Dung T. Nguyen, My T. Thai: Cheap, easy, and massively effective viral marketing in social networks: truth or fiction? HT 2012: 165-174 • Wei Zhang, Weili Wu, Feng Wang, Kuai Xu: Positive influence dominating sets in power-law graphs. Social Netw. Analys. Mining 2(1): 31-37 (2012) • Lidan Fan, Weili Wu, Kai Xing, Wonjun Lee, Ding-Zhu Du, Precautionary Rumor Containment via Trustworthy People in Social Networks.

31. Xu Zhu, Jieun Yu, Wonjun Lee, Donghyun Kim, Shan Shan, Ding-Zhu Du: New dominating sets in social networks. J. Global Optimization 48(4): 633-642 (2010) • Thang N. Dinh, Yilin Shen, Dung T. Nguyen, My T. Thai: On the approximability of positive influence dominating set in social networks. J. Comb. Optim. 27(3): 487-503 (2014)

32. Wei Zhang, Weili Wu, Feng Wang, Kuai Xu: Positive influence dominating sets in power-law graphs. Social Netw. Analys. Mining 2(1): 31-37 (2012) • Theorem. For 2<α<3, there is a polynomial-time constant-approximation.

33. Multi-steps • Given a network, • Find a minimum subset of nodes K which positively dominates all nodes within at most d steps.

34. Thang N. Dinh, Dung T. Nguyen, My T. Thai: Cheap, easy, and massively effective viral marketing in social networks: truth or fiction? HT 2012: 165-174 Theorem

35. THANK YOU! END