On Bharathi-Kempe-Salek Conjecture about Influence Maximization

Ding-Zhu DuUniversity of Texas at Dallas

Outline• Influence Max• BKS-conjecture

2

What is Social Network? Wikipedia Definition: Social Structure •Nodes: Social actors (individuals or organizations)•Links: Social relations

3

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

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

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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

• 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

•Given a digraph and k>0,

•Find k seeds (Kates) to maximize the number of influenced persons (possibly in many steps).

Influence Maximization

8

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Modularity of Influence

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Diffusion Model

• Deterministic diffusion model • Independent Cascade (IC) • Linear Threshold (LT)

12

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).

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

IC Model

• Each person can tell only one person at each moment.

• However, each person may hear from many persons.

15

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

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,

Example

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Stop!

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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 .

Outline• Influence Max• BKS-Conjecture

20

Bharathi-Kempe-Salek Conjecture

21

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Diffusion Model

• Deterministic diffusion model -polynomial-time.

• Linear Threshold (LT) – polynomial-time.• Independent Cascade (IC) – PTAS

22

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

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

1

3

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2

4

6

Step 2: 4--5.

Example

04/21/23 25

A Property of Optimal Solution

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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

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,

Example

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Active Node

Threshold

Active neighbors

vw 0.5

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0.5

0.10.4

0.3 0.2

0.6

0.2

Stop!

U

X

Y

A property

36

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Important understanding on IC

Equivalent Networks

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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).

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

IC Model

• Each person can tell only one person at each moment.

• However, each person may hear from many persons.

52

53

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Important understanding on IC

At non-seed v

54

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Another Dynamic Programming

55

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Open Problem

• IC model• Parameterized algorithms with treewidth as

parameter in IC model.

56

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!!!

Polynomial-time Algorithm

58

Primal or incremental method

duality

Primal-dual

Dynamic program

Divide and conquer

greedy

Local ratio

THANK YOU!