Lecture 2 1 Influence Maximization DingZhu Du University

Lecture 2 -1 Influence Maximization Ding-Zhu Du University of Texas at Dallas

What is Social Network? Wikipedia Definition: Social Structure • Nodes: Social actors (individuals or organizations) • Links: Social relations 2

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 3

Kate Middleton effect “Kate Middleton effect The trend effect that Kate, Duchess of Cambridge has on others, from cosmetic surgery for brides, to sales of coral-colored jeans. ” 4

Hike in Sales of Special Products n According to Newsweek, "The Kate Effect may be worth £ 1 billion to the UK fashion industry. " n Tony Di. Masso, L. K. Bennett’s US president, stated in 2012, ". . . when she does wear something, it always seems to go on a waiting list. " 5

How to Find Kate? • 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! 6

Influence Maximization • Given a digraph and k>0, • Find k seeds (Kates) to maximize the number of influenced persons (possibly in many steps). 7

Theorem Proof 8

Modularity of Influence 9

What is a submodular function? Consider a function f on all subsets of a set E. f is submodular if

What is monotone nodecreasing f is monotone nondecreasing if

Property of Decreasing Marginal Value 1 2

Modularity of Influence 13

Theorem 14

Submadular Function Max 15

Greedy Algorithm 16

Performance Ratio Theorem (Nemhauser et al. 1978) Proof 17

Proof Monotone nondecreasing Submodular! Why? 18

Diffusion Model • Deterministic diffusion model • Independent Cascade (IC) • Linear Threshold (LT) 19

Deterministic Model 2 6 1 5 3 4 both 1 and 6 are source nodes. Step 1: 1 --2, 3; 6 --2, 4. . 5/19/2021 20

Example 2 6 1 5 3 4 Step 2: 4 --5. 5/19/2021 21

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.

References 23

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