![Page 1: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/1.jpg)
On Bharathi-Kempe-Salek Conjecture about Influence Maximization
Ding-Zhu DuUniversity of Texas at Dallas
![Page 2: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/2.jpg)
Outline• Influence Max• BKS-conjecture
2
![Page 3: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/3.jpg)
What is Social Network? Wikipedia Definition: Social Structure •Nodes: Social actors (individuals or organizations)•Links: Social relations
3
![Page 4: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/4.jpg)
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 4
![Page 5: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/5.jpg)
The trend effect that Kate, Duchess of Cambridge has on others, from cosmetic surgery for brides, to sales of coral-colored jeans.”
“Kate Middleton effect
Kate Middleton effect
5
![Page 6: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/6.jpg)
According to Newsweek, "The Kate Effect may be worth £1 billion to the UK fashion industry."
Tony DiMasso, L. K. Bennett’s US president, stated in 2012, "...when she does wear something, it always seems to go on a waiting list."
Hike in Sales of Special Products
6
![Page 7: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/7.jpg)
• Influential persons often have many friends.
• Kate is one of the persons that have many friends in this social network.
For more Kates, it’s not as easy as you might think!
How to Find Kate?
7
![Page 8: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/8.jpg)
•Given a digraph and k>0,
•Find k seeds (Kates) to maximize the number of influenced persons (possibly in many steps).
Influence Maximization
8
![Page 9: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/9.jpg)
9
. 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
1S 2S mS
1u nu2u
ji Su
nodes influence seeds solution hasCover -Set knk
![Page 10: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/10.jpg)
Modularity of Influence
10
.submodular and increasing monotone is )(Then
.set seedby influenced nodes of # denote )(Let
A
AA
)()( BABA vv
v
B
A
![Page 11: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/11.jpg)
Theorem
11
Max. Influence
for ion approximat-)1( a isGreedy 1 e
![Page 12: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/12.jpg)
Diffusion Model
• Deterministic diffusion model • Independent Cascade (IC) • Linear Threshold (LT)
12
![Page 13: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/13.jpg)
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 14: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/14.jpg)
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 15: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/15.jpg)
IC Model
• Each person can tell only one person at each moment.
• However, each person may hear from many persons.
15
![Page 16: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/16.jpg)
Linear Threshold (LT) Model• A node v has random threshold ~ U[0,1]• A node v is influenced by each neighbor w according to a
weight bw,v such that
• A node v becomes active when at least
(weighted) fraction of its neighbors are active
v
v
1 ofneighbor , vwvwb
vvw
vwb ofneighbor active
,
![Page 17: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/17.jpg)
Example
Inactive Node
Active Node
Threshold
Active neighbors
vw 0.5
0.30.2
0.5
0.10.4
0.3 0.2
0.6
0.2
Stop!
U
X
Y
![Page 18: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/18.jpg)
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 19: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/19.jpg)
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 20: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/20.jpg)
Outline• Influence Max• BKS-Conjecture
20
![Page 21: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/21.jpg)
Bharathi-Kempe-Salek Conjecture
21
root. a into directed
cearborescenfor hard-NP ison maximizati Influence
311-306 :2007 WINENetworks. Socialin on Maximizati Influence
eCompetitiv :SalekMahyar Kempe, David Bharathi,Shishir
![Page 22: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/22.jpg)
Diffusion Model
• Deterministic diffusion model -polynomial-time.
• Linear Threshold (LT) – polynomial-time.• Independent Cascade (IC) – PTAS
22
![Page 23: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/23.jpg)
Deterministic Diffusion Model
When a node becomes active (infected or protected), it activates all of its currently inactive (not infected and not protected) neighbors.
The activation attempts succeed with a probability 1.
23
![Page 24: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/24.jpg)
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 24
![Page 25: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/25.jpg)
1
3
5
2
4
6
Step 2: 4--5.
Example
04/21/23 25
![Page 26: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/26.jpg)
A Property of Optimal Solution
26
leaf. aat located
be should seedevery solution, optimalIn
vk
.least at
is leaves ofnumber theassume ,simplicityFor
k
![Page 27: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/27.jpg)
27
1u
vk
Naïve Dynamic Programming
vdu
.at rooted cearborescen
in the placed are seeds when nodes
influenced ofnumber maximum theis ),(
. of degree theis
v
k
kvf
vdv
![Page 28: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/28.jpg)
Naïve Dynamic Programming
28
iu
vik
.0 if ,1
,0 if 0,),(
then leaf, a is If
)}.,(),({max1),(
then leaf, anot is If
111
k
kkvf
v
kufkufkvf
v
vvvdddkkk
![Page 29: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/29.jpg)
Running Time
29
. of neighbors-inat rooted subtrees the
toseeds distribute toexamined are sallocation ))((
. of degree-in thedenotes
vd
kdkO
vd
v
dv
v
v
It is not a polynomial-time!
1u
v1k
![Page 30: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/30.jpg)
Counting
30
).)((1
))((1
1 sallocation of #
rooms.different into allocated are persons
kv
v
dv
v
v
v
dkOdk
k
dkOdk
d
dk
v
![Page 31: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/31.jpg)
Virtual Nodes
31
vv
Change arborescence to binary arborescence
At most n virtual nodes can be introduced.
![Page 32: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/32.jpg)
Weight
32
otherwise 1
node virtuala is if 0 uwu
.at rooted cearborescen
in the placed are seeds when nodes
influenced of weight totalmaximum theis ),(
v
k
kvf
![Page 33: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/33.jpg)
Naïve Dynamic Programming
33
1u
v1k
.0 if ,1
,0 if 0,),(
then leaf, a is If
)}.,(),({max),(
then leaf, anot is If
221121
k
kkvf
v
kufkufwkvf
v
kkkv
2u
![Page 34: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/34.jpg)
Linear Threshold (LT) Model• A node v has random threshold ~ U[0,1]• A node v is influenced by each neighbor w according to a
weight bw,v such that
• A node v becomes active when at least
(weighted) fraction of its neighbors are active
v
v
1 ofneighbor , vwvwb
vvw
vwb ofneighbor active
,
![Page 35: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/35.jpg)
Example
Inactive Node
Active Node
Threshold
Active neighbors
vw 0.5
0.30.2
0.5
0.10.4
0.3 0.2
0.6
0.2
Stop!
U
X
Y
![Page 36: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/36.jpg)
A property
36
edges. live ofselection random under the paths,
edge-live via from reachable setsover on distributi The )2(
. from starting completion toprocess ThresholdLinear the
runningby obtained sets activeover on distributi The (1)
:same theare nodes of setsover
onsdistributi twofollowing the,set seedgiven aFor
A
A
A
?)()( @,, AAbp CLTwuwu
![Page 37: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/37.jpg)
@C model
• Influence can be made only through private talk of person to p.erson
37
![Page 38: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/38.jpg)
38
.1
yprobabilit with activenobody makes )1(
.y probabilit with active makes )(
.y probabilit with active makes (1)
:events exclusivemutually possible 1only are
then there,,...,, neighbors-out has node a If
1
11
21
k
k
vuvu
vuk
vu
k
pp
vk
puvk
puv
k
uuukv
Important understanding on IC
![Page 39: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/39.jpg)
Equivalent Networks
39
1p
3p
2p
1p
3p
2p1p
![Page 40: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/40.jpg)
40
.1
yprobabilit with active makesnobody )1(
.y probabilit with active makes )(
.y probabilit with active makes (1)
:events exclusivemutually possible 1only are there
then ,,...,, neighbors (coming) has node a If
1
11
21
vuvu
vuk
vu
k
k
k
pp
vk
pvuk
pvu
k
uuukv
Additional Condition in @C
![Page 41: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/41.jpg)
Equivalent Networks
41
1p
3p
2p1p
3p
2p
1p
![Page 42: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/42.jpg)
A Property of @C
42
. to from paths ofset ),(
alive. being edge ofy probabilit
)(),(
@
vuvuP
ep
pA
e
Vv Au vuPP PeeC
![Page 43: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/43.jpg)
43
x y x y
xyx 1 y1
yxyx
yxyx
11
leaf. aat located be to
neednot may seedeach solution, optimalIn
![Page 44: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/44.jpg)
44
seed)} anot is |,( seed), a is |,(max{
),(
vkvfvkvf
kvf
![Page 45: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/45.jpg)
At seed v
45
1uv
1k2u
)},(),({max1
)seed a is |,(
221121kufkuf
vkvf
kkk
![Page 46: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/46.jpg)
At non-seed v
46
1uv
1k 2u
)},(),({max
active) becomes Pr()seed anot is |,(
221121kufkuf
vwvkvf
kkk
v
![Page 47: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/47.jpg)
At non-seed v
47
1uv
1k
2u
)},1,(),1,({max
)seed anot is |,(
221121kufkuf
vkvf
kkk
![Page 48: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/48.jpg)
At non-seed v
48
1uv
1k
2u
)},1,(),1,({max
)seed anot is |,,(
221121kiufkiuf
vkivf
kkk
![Page 49: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/49.jpg)
At seed v
49
1uv
1k2u
)},(),({max
1
)seed a is |,,(
2211
1211
21
21
kufkuf
wppwppwp
vkivf
kkk
vivv i
1v 2v
1p 2p
![Page 50: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/50.jpg)
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 51: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/51.jpg)
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 52: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/52.jpg)
IC Model
• Each person can tell only one person at each moment.
• However, each person may hear from many persons.
52
![Page 53: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/53.jpg)
53
.1
yprobabilit with activenobody makes )1(
.y probabilit with active makes )(
.y probabilit with active makes (1)
:events exclusivemutually possible 1only are
then there,,...,, neighbors-out has node a If
1
11
21
k
k
vuvu
vuk
vu
k
pp
vk
puvk
puv
k
uuukv
Important understanding on IC
![Page 54: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/54.jpg)
At non-seed v
54
1uv
ik1 iu
)},(),({max
active) becomes Pr()seed anot is |,(
221121kufkuf
vwvkvf
kkk
v
![Page 55: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/55.jpg)
Another Dynamic Programming
55
active) becomes Pr(
active) becomes Pr( where
} ),,(),,({max
),,(
22
11
21222111)1)(1(1 21
21
uq
uq
qqqkufqkuf
qkvf
iqqq
kkk
.active) Pr( condition under ),(),,( qvkvfqkvf
? of valuespossiblemany How q
![Page 56: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/56.jpg)
Open Problem
• IC model• Parameterized algorithms with treewidth as
parameter in IC model.
56
![Page 57: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/57.jpg)
Bharathi-Kempe-Salek Conjecture
57
root. a into directed cearborescenfor
hard-NP is model ICon with maximizati Influence
311-306 :2007 WINENetworks. Socialin on Maximizati Influence
eCompetitiv :SalekMahyar Kempe, David Bharathi,Shishir
Open!!!
![Page 58: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/58.jpg)
Polynomial-time Algorithm
58
Primal or incremental method
duality
Primal-dual
Dynamic program
Divide and conquer
greedy
Local ratio
![Page 59: On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University of Texas at Dallas](https://reader035.vdocuments.us/reader035/viewer/2022081513/5697bfcf1a28abf838ca9b71/html5/thumbnails/59.jpg)
THANK YOU!