1. Positive Influence Dominating Set
Lecture 5-1
Positive Influence Dominating Set
Ding-Zhu Du
Univ of Texas at Dallas
First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies. 
It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on..
The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups.
Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

2. Social Networks >1000 million users
The 3rd largest “Country” in the world
More visitors than Google
>780 million users
2013, users, 40% yearly increase
400 million
2009, 2 billion tweets per quarter
2010, 4 billion tweets per quarter
2011, tweets per quarter
Facebook Now Has more than one Billion Monthly Active Users
Tencent QQ, popularly known as QQ, is an instant messaging software service QQ also offers a variety of services, including online social games, music, shopping, microblogging, and group and voice chat.
As of 20 March 2013, there are million active QQ accounts, with a peak of million simultaneous online QQ users.[2]
Pinterest is a tool for collecting and organizing things you love.
Facebook , Pinterest, and teitter are dominating.
25 billion
More than 6 billion images
Pinterest, with a traffic higher than Twitter and Google

3. A Trillion Dollar Opportunity
Online Social networks have become a bridge to connect our daily physical life and the virtual web space
On2Off Commerce[1]
With the growth of local commerce on the Web, the links between online and physical commerce are becoming stronger.
Think about the succefssful websites such as Groupon, priceline, OpenTable,Restaurant.com, and SpaFinder etc. What do they have in common? They grease the wheels of online-to-offline commerce.
The key to O2O is that it finds consumers online and brings them into real-world stores. It has been predicted that it will create a trillion dollar opportunity via on 2 off e commerce.
So online social networks provide good platforms for whose who want to move their physical business to online business.
[Venture capitalists and entrepreneurs would be wise to think beyond cloning the “deal of the day” concept—and instead think about how the discovery, payment, and performance measurement of offline commerce can move online.]
[1] Online to Offline is trillion dollar business

4. “Love Obama” I hate Obama, the worst president ever I love Obama
Obama is fantastic
Obama is great!
No Obama in 2012!
A fundamental understanding of communication has always been at the center of a politician's arsenal, but a firm grasp on the future of communication can be the secret weapon that wins the war. For Franklin D. Roosevelt, it was radio. For John F. Kennedy, it was television. And for Barack Obama, it is social media.
The 2008 Obama Presidential campaign made history.  Not only was Obama the first African American to be elected president, but he was also the first presidential candidate to effectively use social media as a major campaign strategy.
During his campaign, Obama has made his Web 2.0 presence known. He has over 1.5 million friends on MySpace and Facebook, and he currently has over 45,000 followers on Twitter. This personal activity in social networks allows him to quickly get the word out across multiple platforms.
He cannot be the next president!
Positive
Negative

5. What is Social Influence?
Social influence occurs when one's opinions, emotions, or behaviors are affected by others, intentionally or unintentionally.[1]
Informational social influence: to accept information from another;
Normative social influence: to conform to the positive expectations of others.
[1]

6. Three Degree of Influence
Six degree of separation[1]
Three degree of Influence[2]
Michael Gurevich conducted seminal work in his empirical study of the structure of social networks in 1961, everyone and everything is six or fewer steps away.
Christakis and Fowler found that social networks have great influence on individuals' behavior. But social influence does not end with the people to whom a person is directly tied. our actions can influence people we have never met. The influence ceases to have a noticeable effect on people beyond three degrees of separation.
150*150*150=1,375,000
You are able to influence up to over 1,000,000 persons in the world, according to the Dunbar’s number[3].
[1] S. Milgram. The Small World Problem. Psychology Today, 1967, Vol. 2, 60–67
[2] J.H. Fowler and N.A. Christakis. The Dynamic Spread of Happiness in a Large Social Network: Longitudinal Analysis Over 20 Years in the Framingham Heart Study. British Medical Journal 2008; 337: a2338
[3] R. Dunbar. Neocortex size as a constraint on group size in primates. Human Evolution, 1992, 20: 469–493.

7. Viral Marketing Viral marketing or word of mouth strategy
exploits the social network values of customers
the most effective marketing strategy
To study the effects of “word of mouth” in the promotion of new products, Domingos and Richardson posed an algorithmic problem in the year 2001.
Research shows that customers, when buying a product, trust the review information obtained from their friends far more than advertisements from public media [3]. And it leverages the customers themselves to carry out the promoting effect
Social networks also provides good platforms for viral marketing since it transmits messages quickly.
Research shows that customers, when buying a product, trust the review information obtained from their friends far more than advertisements from public media [3]. And it leverages the customers themselves to carry out the promoting effect.
A consumer tells five to ten people and then those five to ten people tell another five to ten people. The driving force behind most viral campaigns is the passion a consumer carries. It's like a virus that continuously infects more people and spreads without requiring anymore marketing effort. .

8. Illustrative Example :Influence Maximization
A new product is available in the market.
Whom to give free samples to maximize the purchase of the product ?
To study the effects of “word of mouth” in the promotion of new products, Domingos and Richardson posed an algorithmic problem in the year 2001.

9. Problem Setting Given Goal Question
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
if we can try to convince a subset of individuals to adopt a new
product or innovation
But how should we choose the few key individuals to use for seeding this process?
Which blogs should one read to be most up to date?

10. Product Advertisement

11. More Companies: Competition

12. How to Measure Influence?

13. A Winning Strategy: Positive Influence

14. Positive Influence is a Special Threshold Model
This is not submodular! B x A

15. Positive-Influence MaxB x A

16. Min Positive-influence Dominating Set
Given a network,
Find a minimum positive-influence dominating set.

17. Greedy Algorithm

18. Section 2.4

19. A General Problem

20. Greedy Algorithm

21. A General Theorem
Remark:

22. Xu Zhu, Jieun Yu, Wonjun Lee, Donghyun Kim, Shan Shan, Ding-Zhu Du: New dominating sets in social networks. J. Global Optimization 48(4): (2010)
Min Positive Dominating Set is APX-had.
There exist a polynomial-time (1+ln 1.5Δ)-approximation.

23. Potential Function
Lemma 1
Proof 1 2 3

24. Proof of 1 B v A x

25. Proof of 2

27. Proof of 3

28. Lemma 2(a)
Proof

29. Lemma 2(b)
Proof

30. Greedy Algorithm

31. A General Theorem
Remark:

32. Theorem
Remark:

33. Thang N. Dinh, Yilin Shen, Dung T. Nguyen, My T
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): (2014)
Theorem

34. Open Problem 1

35. Open Problem 2

36. Open Problem 3

37. THANK YOU!