Maximizing the Spread of Maximizing the Spread of I nfluence - - PowerPoint PPT Presentation

Maximizing the Spread of Maximizing the Spread of I nfluence through a Social I nfluence through a Social Network Network

By David Kempe, Jon Kleinberg, By David Kempe, Jon Kleinberg, Eva Tardos Eva Tardos Report by Joe Abrams Report by Joe Abrams

Social Networks Social Networks

Infectious disease networks Infectious disease networks

Viral Marketing Viral Marketing

Viral Marketing Viral Marketing

• Example:

Example: Hotmail Hotmail

• Included service

Included service’ ’s URL in every email sent s URL in every email sent by users by users

• Grew from zero to 12 million users in 18

Grew from zero to 12 million users in 18 months with small advertising budget months with small advertising budget

Domingos and Richardson Domingos and Richardson (2001, 2002) (2001, 2002)

• Introduction to maximization of influence

Introduction to maximization of influence

• ver social networks
• ver social networks
• Intrinsic Value vs. Network Value

Intrinsic Value vs. Network Value

• Expected Lift in Profit (ELP)

Expected Lift in Profit (ELP)

• Epinions,

Epinions, “ “web of trust web of trust” ”, 75,000 users and , 75,000 users and 500,000 edges 500,000 edges

Domingos and Richardson Domingos and Richardson (2001, 2002) (2001, 2002)

• Viral marketing (using greedy hill

Viral marketing (using greedy hill-

• climbing

climbing strategy) worked very well compared with strategy) worked very well compared with direct marketing direct marketing

• Robust (69% of total lift knowing only 5%

Robust (69% of total lift knowing only 5%

• f edges)
• f edges)

Diffusion Model: Linear Diffusion Model: Linear Threshold Model Threshold Model

• Each node (consumer) influenced by set

Each node (consumer) influenced by set

• f neighbors; has threshold
• f neighbors; has threshold Θ

Θ from from uniform distribution [0,1] uniform distribution [0,1]

• When combined influence reaches

When combined influence reaches threshold, node becomes threshold, node becomes “ “active active” ”

• Active node now can influence its

Active node now can influence its neighbors neighbors

• Weighted edges

Weighted edges

Diffusion Model: Linear Diffusion Model: Linear Threshold Model Threshold Model

Diffusion Model: Independent Diffusion Model: Independent Cascade Model Cascade Model

• Each active node has a probability

Each active node has a probability p p of

• f

activating a neighbor activating a neighbor

• At time

At time t t+1, all newly activated nodes try +1, all newly activated nodes try to activate their neighbors to activate their neighbors

• Only one attempt for per node on target

Only one attempt for per node on target

• Akin to turn

Akin to turn-

• based strategy game?

based strategy game?

Influence Maximization Influence Maximization

• Using greedy hill

Using greedy hill-

• climbing strategy, can

climbing strategy, can approximate optimum to within a factor of approximate optimum to within a factor of (1 (1 – – 1/e 1/e – – ε ε), or ~63% ), or ~63%

• Proven using theories of submodular

Proven using theories of submodular functions (diminishing returns) functions (diminishing returns)

• Applies to both diffusion models

Applies to both diffusion models

Testing on network data Testing on network data

• Co

Co-

• authorship network

authorship network

• High

High-

• energy physics theory section of

energy physics theory section of www.arxiv.org www.arxiv.org

• 10,748 nodes (authors) and ~53,000

10,748 nodes (authors) and ~53,000 edges edges

• Multiple co

Multiple co-

• authored papers listed as

authored papers listed as parallel edges (greater weight) parallel edges (greater weight)

Testing on network data Testing on network data

• Linear Threshold: influence weighed by #

Linear Threshold: influence weighed by #

• f parallel lines, inversely weighed by
• f parallel lines, inversely weighed by

degree of target node: w = c degree of target node: w = cu,v

u,v /d

/dv

v

• Independent Cascade:

Independent Cascade: p p set at 1% and set at 1% and 10%; total probability for 10%; total probability for u v u v is is 1 1 – – (1 (1 – – p p)^c )^cu,v

u,v

• Weighted Cascade:

Weighted Cascade: p p = 1/ d = 1/ dv

v

Algorithms Algorithms

• Greedy hill

Greedy hill-

• climbing

climbing

• High degree: nodes with greatest number

High degree: nodes with greatest number

• f edges
• f edges
• Distance centrality: lowest average

Distance centrality: lowest average distance with other nodes distance with other nodes

• Random

Random

Algorithms Algorithms

Results: Linear Threshold Model Results: Linear Threshold Model

Greedy: ~40% better than central, ~18% better than high degree

Results: Weighted Cascade Results: Weighted Cascade Model Model

Results: Independent Cascade, Results: Independent Cascade, p p = 1% = 1%

Results: Independent Cascade, Results: Independent Cascade, p p = 10% = 10%

Advantages of Random Selection Advantages of Random Selection

Generalized models Generalized models

• Generalized Linear Threshold: for node

Generalized Linear Threshold: for node v v, , influence of neighbors not necessarily sum influence of neighbors not necessarily sum

• f individual influences
• f individual influences
• Generalized Independent Cascade: for

Generalized Independent Cascade: for node node v v, probability , probability p p depends on set of depends on set of v v’ ’s s neighbors that have previously tried to neighbors that have previously tried to activate activate v v

• Models computationally equivalent,

Models computationally equivalent, impossible to guarantee approximation impossible to guarantee approximation

Non Non-

• Progressive Threshold

Progressive Threshold Model Model

• Active nodes can become inactive

Active nodes can become inactive

• Similar concept: at each time

Similar concept: at each time t t, whether , whether

• r not
• r not v

v becomes/stays active depends on becomes/stays active depends on if influence meets threshold if influence meets threshold

• Can

Can “ “intervene intervene” ” at different times; need at different times; need not perform all interventions at not perform all interventions at t t = 0 = 0

• Answer to progressive model with graph G

Answer to progressive model with graph G equivalent to non equivalent to non-

• progressive model with

progressive model with layered graph G layered graph Gτ

τ

General Marketing Strategies General Marketing Strategies

• Can divide up total budget

Can divide up total budget κ κ into equal into equal increments of size increments of size δ δ

• For greedy hill

For greedy hill-

• climbing strategy, can

climbing strategy, can guarantee performance within factor of guarantee performance within factor of 1 1 – – e^[ e^[-

• (

(κ κ * *γ γ)/( )/(κ κ + + δ δ * *n n)] )]

• As

As δ δ decreases relative to decreases relative to κ κ, result , result approaches 1 approaches 1 – – e e-

• 1

1 = 63%

= 63%

Strengths of paper Strengths of paper

• Showed results in two complementary

Showed results in two complementary fashions: theoretical models and test fashions: theoretical models and test results using real dataset results using real dataset

• Demonstrated that greedy hill

Demonstrated that greedy hill-

• climbing

climbing strategy could guarantee results within strategy could guarantee results within 63% of optimum 63% of optimum

• Used specific and generalized versions of

Used specific and generalized versions of two different diffusion models two different diffusion models

Weaknesses of paper Weaknesses of paper

• Doesn

Doesn’ ’t fully explain methodology of t fully explain methodology of greedy hill greedy hill-

• climbing strategy

climbing strategy

• Lots of work not shown

Lots of work not shown – – simply refers to simply refers to work done in other papers work done in other papers

• Threshold value uniformly distributed?

Threshold value uniformly distributed?

• Influence inversely weighted by degree of

Influence inversely weighted by degree of target? target?

Questions? Questions?