Lecture 5-1 Positive Influence Dominating Set Ding-Zhu Du Univ of Texas at Dallas

Social Networks >1000 million users The 3 rd largest “Country” in the world More visitors than Google More than 6 billion images 2009, 2 billion tweets per quarter 2010, 4 billion tweets per quarter 2011, tweets per quarter >780 million users Pinterest, with a traffic higher than Twitter and Google 25 billion 2013, users, 40 % yearly increase 400 million 2

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] [1] Online to Offline is trillion dollar business 3

“Love Obama” I love Obama Obama is great! Obama is fantastic I hate Obama, the worst president ever He cannot be the next president! No Obama in 2012! Positive Negative 4

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] 5

Three Degree of Influence Three degree of Influence [2] [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. Six degree of separation [1] You are able to influence up to over 1,000,000 persons in the world, according to the Dunbar’s number [3]. 6

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.

Illustrative Example :Influence Maximization A new product is available in the market. Whom to give free samples to maximize the purchase of the product ? 8

Problem Setting 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 9

Product Advertisement 10

More Companies: Competition 11

How to Measure Influence? 12

A Winning Strategy: Positive Influence 13

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

Positive-Influence Max 15 A B x

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

Greedy Algorithm

Section

A General Problem

Greedy Algorithm

A General Theorem Remark:

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) Xu ZhuJieun YuWonjun LeeDonghyun Kim Shan J. Global Optimization Min Positive Dominating Set is APX-had. There exist a polynomial-time (1+ln 1.5Δ)- approximation.

Potential Function 23 Lemma 1 Proof 1 2 3

Proof of 24 1 A B x v

Proof of 25 2

26

Proof of 27 3

Lemma 2(a) 28 Proof

Lemma 2(b) 29 Proof

Greedy Algorithm

A General Theorem Remark:

Theorem

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) Thang N. DinhYilin ShenDung T. NguyenJ. Comb. Optim Theorem

Open Problem 1 34

Open Problem 2 35

Open Problem 3 36

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