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Lecture 3-1
Independent Cascade
Weili Wu Ding-Zhu DuUniversity of Texas at Dallas
Outline• Influence Max• Independent Cascade
2
•Given a digraph and k>0,
•Find k seeds (Kates) to maximize the number of influenced persons.
Influence Maximization
3
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
Modularity of Influence
5
.submodular and increasing monotone is )(Then
.set seedby influenced nodes of # denote )(Let
A
AA
)()( BABA vv
v
B
A
Submadular Function Max
6
kS
VS
Sf
Rf V
||
subject to
)( max
Consider
function. submodular increasing
monotone nonegative a be 2:Let
Greedy Algorithm
7
.output
};{
set and )( maximize to choose
do to1for
;
1
1
0
k
ii
iv
S
vSS
SfVv
ki
S
Performance Ratio
8
solutionm. optimalan is * where
*)()1()( 1
S
SfeSf k
Theorem (Nemhauser et al. 1978)
Proof
Theorem
9
Max. Influence
for ion approximat-)1( a isGreedy 11 e
Outline• Influence Max• Independent Cascade
10
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
1
3
5
2
4
6
Step 2: 4--5.
Example
04/21/23 12
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.
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).
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
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
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.
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 .
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
20
)()(
|}in path
directed a via from reached becan |{|)( :Note
.submodular monotone is )( .
BABA
X
AuuA
AClaim
Xv
Xv
X
X
v
B
A
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?
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
23
G
ts
G
ts 't
'G
})({})({),,( ' ssGtsp GG
1', ttpGvup vu in ),(for 2/1,
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?
Monte-Carlo Method
25
Buffon's needle
tp
2
)(#
2
when
cross
n
t
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'
Editor-in-Chief: Ding-Zhu Du My T. Thai
Computational Social Networks
27
A New Springer Journal
Welcome to Submit Papers
THANK YOU!
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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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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• 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
31
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)
32
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
33