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

2. Product Advertisement 2

3. More Companies: Competition 3

4. How to Measure Influence? 4

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

6. Positive-Influence Max 6 A B x

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 7

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

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

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.) 10

11. Potential Function 11 Lemma 1 Proof 1 2 3

12. Greedy Algorithm

13. Theorem

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

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

20. 20

21. 21

22. Theorem 22

23. 23 Lemma 2 Proof Lemma 1

24. 24 Lemma 3 Proof

25. Theorem 25 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. 26

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

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? 28

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

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)Hongwei DuErika CamachoKuai Xu Wonjun LeeYan ShiShan Theor. Comput. Sci. 412 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 Thang N. DinhDung T. NguyenHT 2012 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) Wei ZhangFeng WangKuai Xu Social Netw. Analys. Mining 2 Lidan Fan, Weili Wu, Kai Xing, Wonjun Lee, Ding-Zhu Du, Precautionary Rumor Containment via Trustworthy People in Social Networks. 30

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) Xu ZhuJieun YuWonjun LeeDonghyun Kim Shan J. Global Optimization 48 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) Thang N. DinhYilin ShenDung T. NguyenJ. Comb. Optim. 27 31

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) Wei ZhangFeng WangKuai XuSocial Netw. Analys. Mining 2 Theorem. For 2<α<3, there is a polynomial- time constant-approximation. 32

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

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 Thang N. DinhDung T. NguyenHT 2012 34 Theorem

35. THANK YOU! END