lecture 5-1 positive influence dominating set ding-zhu du univ of texas at dallas
DESCRIPTION
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 3TRANSCRIPT
Social Networks• >1000 million users• The 3rd 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 increase400 million
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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 businesshttp://techcrunch.com/2010/08/07/why-online2offline-commerce-is-a-trillion-dollar-opportunity/
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 Negative4
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] http://en.wikipedia.org/wiki/Social_influence 5
Three Degree of InfluenceThree 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
A Winning Strategy:Positive Influence
13
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Positive Influence is a Special Threshold Model
14
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5.0|)(|/|| if 05.0|)(|/|| if 1
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Positive-Influence Max
15
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)(10)( BfAf xx
Min Positive-influence Dominating Set
• Given a network,• Find a minimum positive-influence dominating
set.
16
Greedy Algorithm
.output };{
and )( maximize to choosedo )()( while
;
AxAA
AgAVxVgAg
A
x
Section 2.4
18
A General ProblemA General Problem
)( s.t.
)()( min
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submodular ,increasing monotone aConsider
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do )(* while;
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A General TheoremA General Theorem
}).({max whereionapproximat-)ln1( a givesGreedy then ,for
0)( and function,integer an is ,0)( If
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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): 633-642 (2010)
22
• Min Positive Dominating Set is APX-had.• There exist a polynomial-time (1+ln 1.5Δ)-
approximation.
Potential Function
23
.0)( and submodular monotone is )(
.))(deg, 2/)deg(min()(
|}or ,),(|{|)(deg , and For ).,(graph aConsider
gg
vvAg
AvAuEvuuvVAVvEVG
VvA
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Lemma 1
vv
vv
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AAg
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).()( is sothen
,0)( and submodular monotone is every If.0)( and submodular
monotone is ))(,min((A) ,0constant any For .0)( and submodular monotone is )(deg)(
Proof1
2
3
Proof of
24
1
A
B
x
)(deg)(deg)(deg)(deg }{}{ vvvv BxBAxA
v
Proof of
25
2
))(,min()( AfcA
. Assume BA
.0))(,min(0)()).(,min())(,min()()(
fcfBfcAfcBfAf
)(}){()( )()(}){(
.}){( 1. Case
BxBBcAcAxA
cxAf
26
))(,min()()(}){()(}){(
AfcABfxBfAfxAf
)(}){( )(}){(
)(}){()(}){(.)( and }){( 2. Case
BxBBfxBf
AfxAfAxAcBfcxAf
)(}){( )(}){(
)(}){()(}){(.)( and }){( 3. Case
BxBccBfxBf
AfxAfAxAcBfcxAf
Proof of
27
3
)(}){()(}){( BxBAxA vvvv
v
vvv
vv BxBAxA )(}){()(}){(
0)(0)( v
vv
Lemma 2(a)
28
).()(set dominating influence-positivea is VgAgA
Proof
)(2/)deg()(
2/)deg())(deg,2/)deg(min( , For
set dominating influence-positivea is
VgvAg
vvvAVv
A
Vv
A
Lemma 2(b)
29
).(}){(:)()( AgxAgAVxVgAg
Proof
).(}){(
))(deg,2/)deg((min 2/)deg(
))(deg,2/)deg(min( )(deg2/)deg(:)()(
}{
AgvAgvv
v
vvvvAVvVgAg
A
vA
A
Greedy Algorithm
.output };{
and )( maximize to choosedo )()( while
;
AxAA
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A
x
A General TheoremA General Theorem
}).({max whereionapproximat-)ln1( a givesGreedy then ,for
0)( and function,integer an is ,0)( If
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)).(})({(max Therefore ).()()(consider may we,0)( If
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f
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Remark:
TheoremTheorem
.degree node maximum theis where
ionapproximat-)23ln1(a givesGreedy
Remark:
.23)deg(
23max
))deg(21)(deg(max
})({max
v
vv
vg
Vv
Vv
Vv
• 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)
33
. unless problemSet Dominating PositiveMin for ion approximat-)lnlnln(
time-polynomial no exists there,0constant someFor
PNPΔ ΔΔ-
c
Theorem
Open Problem 1
34
steps. severalafter influenced positively be willnodeevery such that set seed
minimum a find ),,(network social aGiven A
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Open Problem 2
35
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positive a and ),(network social aGiven
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Open Problem 3
36
steps. someafter by by influenced positively nodes of # the
maximize tonode of set seed a find ,integer positive a and ),(network social aGiven
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