Influence Maximization 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 a social structure made of social actors (individuals or organizations) called “nodes”, which are tied by one or more specific Social relations, such as friendship, kinship, religions, knowledge or prestige. The nodes or actors as part of network data would seem to be pretty straight-forward. Our study mainly focus on the relations among actors.

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

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.” “Kate Middleton effect Duchess of Cambridge, Kate, is a fashion icon who leads in fashion circles. she seems to become more and more beautiful after she married Prince William. Many articles referred her nose as “adorable,” “perky,” and “feminine.” Women in the New York and Long Island areas are rushing out to seek a plastic surgeon. There are approximately 100 young women a month so far that are requesting cosmetic surgery on their nose to get an exact duplicate of Kate’s.

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." The fashion choices of the Duchess of Cambridge, Kate Middleton, have already brought $1 billion into the British economy. Reiss, the High Street store that Kate loves, is planning to open 13 outposts in the U.S. within this year.

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). Market salesman

Theorem Proof

Modularity of Influence

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

Property of Decreasing Marginal Value 1 2

Modularity of Influence

Theorem

Submadular Function Max

Greedy Algorithm

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 Two kinds of influence cascades: rumors and protectors. Each individual has three status: inactive, rumored, protected. The active individual activates all of its neighbors successfully. When rumors and protectors influence an individual at the same time, then the individual is protected. Each individual only has one chance to influence their neighbors. A node will never change its status if it has been activated. 3 4 both 1 and 6 are source nodes. Step 1: 1--2,3; 6--2,4. . 7/17/2018 20

Example 6 2 1 5 3 4 Step 2: 4--5. 7/17/2018

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. the influence of a set of nodes A: the expected number of active nodes at the end of the process.

References

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