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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 r elations. What is Social Influence?.
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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
What is Social Influence? [1] http://en.wikipedia.org/wiki/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.
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.”
Hike in Sales of Special Products • 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."
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!
Influence Maximization • Given a digraph and • k>0, • Find k seeds (Kates) • to maximize the number of influenced persons (possibly in many steps).
Theorem Proof
What is a submodular function? Consider a function f on all subsets of a set E. f is submodular if
What is monotone increasing f is monotone increasing if
Performance Ratio Theorem (Nemhauser et al. 1978) Proof
Proof Monotone increasing Submodular! Why?
Diffusion Model • Deterministic diffusion model • Independent Cascade (IC) • Linear Threshold (LT)
Deterministic Model 6 2 1 5 3 4 both 1 and 6 are source nodes. Step 1: 1--2,3; 6--2,4. .
Example 6 2 1 5 3 4 Step 2: 4--5.
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.