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Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

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Page 1: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Lecture 3-1

Independent Cascade

Weili Wu Ding-Zhu DuUniversity of Texas at Dallas

Page 2: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Outline• Influence Max• Independent Cascade

2

Page 3: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

•Given a digraph and k>0,

•Find k seeds (Kates) to maximize the number of influenced persons.

Influence Maximization

3

Page 4: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

4

. toequal isunion whosesubsets theof find

,0integer and},,...,{set ground a of

,..., subsets of collection aGiven :Cover-Set

Max InfluenceCover-Set

hard.- isMax Influence

1

1

Uk

kuuU

SS

NP

n

m

pm

Theorem

Proof

1S2S mS

1unu

2u

ji Su

nodes influence seeds solution hasCover -Set knk

Page 5: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Modularity of Influence

5

.submodular and increasing monotone is )(Then

.set seedby influenced nodes of # denote )(Let

A

AA

)()( BABA vv

v

B

A

Page 6: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Submadular Function Max

6

kS

VS

Sf

Rf V

||

subject to

)( max

Consider

function. submodular increasing

monotone nonegative a be 2:Let

Page 7: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Greedy Algorithm

7

.output

};{

set and )( maximize to choose

do to1for

;

1

1

0

k

ii

iv

S

vSS

SfVv

ki

S

Page 8: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Performance Ratio

8

solutionm. optimalan is * where

*)()1()( 1

S

SfeSf k

Theorem (Nemhauser et al. 1978)

Proof

Page 9: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Theorem

9

Max. Influence

for ion approximat-)1( a isGreedy 11 e

Page 10: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Outline• Influence Max• Independent Cascade

10

Page 11: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Deterministic Model

1

3

4

5

26

both 1 and 6 are source nodes.

Step 1: 1--2,3; 6--2,4. .

04/21/23 11

Page 12: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

1

3

5

2

4

6

Step 2: 4--5.

Example

04/21/23 12

Page 13: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Models of Influence Diffusion• Two basic classes of probabilistic diffusion models:

– threshold and cascade• General operational view:

– A social network is represented as a directed graph, with each person (customer) as a node.

– Nodes start either active or inactive.– An active node may trigger activation of neighboring nodes – Monotonicity assumption: active nodes never deactivate.

Page 14: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Independent Cascade (IC) Model

• When node v becomes active, it has a single chance of activating each currently inactive neighbor w.

• The activation attempt succeeds with probability pvw .

• The deterministic model is a special case of IC model. In this case, pvw =1 for all (v,w).

Page 15: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

15

.1

yprobabilit with activenobody makes )1(

.y probabilit with active makes )(

.y probabilit with active makes (1)

:events possible 1only are then there

,,...,, neighbors has node a If

1

11

21

k

k

vuvu

vuk

vu

k

pp

vk

puvk

puv

k

uuukv

Important understanding

Page 16: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Example

vw 0.5

0.3 0.20.5

0.10.4

0.3 0.2

0.6

0.2

Inactive Node

Active Node

Newly active node

Successful attempt

Unsuccessfulattempt

Stop!

UX

Y

Page 17: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Influence Maximization Problem

• Influence spread of node set S: σ(S) – expected number of active nodes at the end of

diffusion process, if set S is the initial active set.

• Problem Definition (by Kempe et al., 2003): (Influence Maximization). Given a directed and edge-weighted social graph G = (V,E, p) , a diffusion model m, and an integer k ≤ |V |, find a set S V ⊆ , |S| = k, such that the expected influence spread σm(S) is maximum.

Page 18: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Known Results• Bad news: NP-hard optimization problem for both IC and LT

models.• Good news: • σm(S) is monotone and submodular.• We can use Greedy algorithm!

• Theorem: The resulting set S activates at least (1-1/e) (>63%) of the number of nodes that any size-k set could activate .

Page 19: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Proof of Submodularity

19

)()(

|}in path

directed a via from reached becan |{|)( :Note

.submodular monotone is )( .

).(]Pr[)(Then

.set seedfor nodes active ofnumber thedenote

)(let , sampleeach For .digraph input of

subgraph all of consisting space sample aConsider

BABA

X

AuuA

AClaim

AXA

A

AXG

Xv

Xv

X

X

X

X

X

Page 20: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

20

)()(

|}in path

directed a via from reached becan |{|)( :Note

.submodular monotone is )( .

BABA

X

AuuA

AClaim

Xv

Xv

X

X

v

B

A

Page 21: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Decision Version of InfMax in IC

21

models LT and ICfor hard- isInfMax

hard.- is ICin InfMax ofersion Decision v

.)(such that

nodes of subset a exists here whether tdetermine

,0 and 0 integers two),,( edgeeach for

y probabilit active with ),(digraph aGiven

,

NP

NP

hA

kA

hkwup

EVG

IC

wu

Theorem

Corollary

Is it in NP?

Page 22: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

22

LT.or ICfor hard-# is )( Computing mPSmTheorem (Chen et al., 2010)

Proof

).( Computing toreducible Turing

time-polynomial is problem complete-#A

S

P

m

. to frompath a containing of

subgraphsmany howcount , and nodes two

and digraph aGiven :problem complete-#

tsG

ts

GP

).,,(by denoted , toconnected be to

ofy probabilit thecompute toequivalent isit then

connected, be to1/2 ofy probabilit has edgeeach If

Gtspts

Page 23: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

23

G

ts

G

ts 't

'G

})({})({),,( ' ssGtsp GG

1', ttpGvup vu in ),(for 2/1,

Page 24: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Disadvantage• Lack of efficiency.

– Computing σm(S) is # P-hard under both IC and LT models.

– Selecting a new vertex u that provides the largest marginal gain σm(S+u) - σm(S), which can only be approximated by Monte-Carlo simulations (10,000 trials).

• Assume a weighted social graph as input.– How to learn influence probabilities from history?

Page 25: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Monte-Carlo Method

25

Buffon's needle

tp

2

)(#

2

when

cross

n

t

Page 26: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

References

26

(2014) 4(1)

on.maximizati influence social online oapproach t

novelA :Chen Zhiming Wu, WeiliLu, Zaixin Xu, Wen 2.

146.-137 pp. ,2003 network, social a through influence

of spread theMaximizing Tardos, E. Kleinberg, J. Kempe, D. 1.

ningAnalys. Mi

w. Social Net

KDD'

Page 27: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Editor-in-Chief: Ding-Zhu Du My T. Thai

Computational Social Networks

27

A New Springer Journal

Welcome to Submit Papers

Page 28: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

THANK YOU!

Page 29: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Yuqing Zhu, Zaixin Lu, Yuanjun Bi, Weili Wu, Yiwei Jiang, Deying Li: Influence and Profit: Two Sides of the Coin. ICDM 2013: 1301-1306

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Page 30: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Lidan Fan, Zaixin Lu, Weili Wu, Yuanjun Bi, Ailian Wang: A New Model for Product Adoption over Social Networks. COCOON 2013: 737-746

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Page 31: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

• Songsong Li, Yuqing Zhu, Deying Li, Donghyun Kim, Huan Ma, Hejiao Huang: Influence maximization in social networks with user attitude modification. ICC 2014: 3913-3918

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Page 32: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Zaixin Lu, Lidan Fan, Weili Wu, Bhavani Thuraisingham and Kai Yang, Efficient influence spread estimation for influence maximization under the linear threshold model, Computational Social Networks, 1 (2014)

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Page 33: Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Zaixin Lu, Wei Zhang, Weili Wu, Bin Fu, Ding-Zhu Du: Approximation and Inapproximation for the Influence Maximization Problem in Social Networks under Deterministic Linear Threshold Model. ICDCS Workshops 2011: 160-165

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