On Bharathi-Kempe-Salek Conjecture about Influence Maximization Ding-Zhu Du University 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] 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 5

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

9 Theorem Proof

Modularity of Influence 10

Theorem 11

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

Example v w Inactive Node Active Node Newly active node Successful attempt Unsuccessful attempt Stop! U X 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 b w,v such that A node v becomes active when at least (weighted) fraction of its neighbors are active

Example Inactive Node Active Node Threshold Active neighbors v w Stop! U X 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 σ m (S) is maximum.

Known Results Bad news: NP-hard optimization problem for both IC and LT models. Good news: σ m (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

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 both 1 and 6 are source nodes. Step 1: 1--2,3; 6--2,4.. 12/19/201524

Step 2: Example 12/19/201525

A Property of Optimal Solution 26

27 Naïve Dynamic Programming

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Running Time 29 It is not a polynomial-time!

Counting 30

Virtual Nodes 31 Change arborescence to binary arborescence At most n virtual nodes can be introduced.

Weight 32

Naïve Dynamic Programming 33

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 b w,v such that A node v becomes active when at least (weighted) fraction of its neighbors are active

Example Inactive Node Active Node Threshold Active neighbors v w Stop! U X Y

A property 36

@C model Influence can be made only through private talk of person to p.erson 37

38 Important understanding on IC

Equivalent Networks 39

40 Additional Condition

Equivalent Networks 41

A Property 42

43

44

At seed v 45

At non-seed v 46

At non-seed v 47

At non-seed v 48

At seed v 49

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

Example v w Inactive Node Active Node Newly active node Successful attempt Unsuccessful attempt Stop! U X Y

IC Model Each person can tell only one person at each moment. However, each person may hear from many persons. 52

53 Important understanding on IC

At non-seed v 54

Another Dynamic Programming 55

Open Problem IC model Parameterized algorithms with treewidth as parameter in IC model. 56

Bharathi-Kempe-Salek Conjecture 57 Open!!!

Polynomial-time Algorithm 58 Primal or incremental method duality Primal-dual Dynamic program Divide and conquer greedy Local ratio

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