maximizing the spread of influence through a social network by david kempe, jon kleinberg, eva...
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Maximizing the Spread Maximizing the Spread of Influence through a of Influence through a
SocialSocialNetworkNetwork
By David Kempe, Jon By David Kempe, Jon Kleinberg, Eva TardosKleinberg, Eva Tardos
Report by Joe AbramsReport by Joe Abrams
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Social NetworksSocial Networks
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Infectious disease networksInfectious disease networks
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Viral MarketingViral Marketing
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Viral MarketingViral Marketing
• Example:Example: HotmailHotmail
• Included service’s URL in every email sent Included service’s URL in every email sent by usersby users
• Grew from zero to 12 million users in 18 Grew from zero to 12 million users in 18 months with small advertising budgetmonths with small advertising budget
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Domingos and Richardson Domingos and Richardson (2001, 2002)(2001, 2002)
• Introduction to maximization of Introduction to maximization of influence over social networksinfluence over social networks
• Intrinsic Value vs. Network ValueIntrinsic Value vs. Network Value
• Expected Lift in Profit (ELP)Expected Lift in Profit (ELP)
• Epinions, “web of trust”, 75,000 Epinions, “web of trust”, 75,000 users and 500,000 edgesusers and 500,000 edges
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Domingos and Richardson Domingos and Richardson (2001, 2002)(2001, 2002)
• Viral marketing (using greedy hill-Viral marketing (using greedy hill-climbing strategy) worked very well climbing strategy) worked very well compared with direct marketingcompared with direct marketing
• Robust (69% of total lift knowing only Robust (69% of total lift knowing only 5% of edges)5% of edges)
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Diffusion Model: Linear Diffusion Model: Linear Threshold ModelThreshold Model
• Each node (consumer) influenced by Each node (consumer) influenced by set of neighbors; has threshold set of neighbors; has threshold ΘΘ from uniform distribution [0,1]from uniform distribution [0,1]
• When combined influence reaches When combined influence reaches threshold, node becomes “active”threshold, node becomes “active”
• Active node now can influence its Active node now can influence its neighborsneighbors
• Weighted edgesWeighted edges
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Diffusion Model: Linear Diffusion Model: Linear Threshold ModelThreshold Model
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Diffusion Model: Diffusion Model: Independent Cascade ModelIndependent Cascade Model
• Each active node has a probability Each active node has a probability pp of activating a neighborof activating a neighbor
• At time At time tt+1, all newly activated +1, all newly activated nodes try to activate their neighborsnodes try to activate their neighbors
• Only one attempt for per node on Only one attempt for per node on targettarget
• Akin to turn-based strategy game?Akin to turn-based strategy game?
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Influence MaximizationInfluence Maximization
• Using greedy hill-climbing strategy, Using greedy hill-climbing strategy, can approximate optimum to within a can approximate optimum to within a factor of (1 – 1/e – factor of (1 – 1/e – εε), or ~63%), or ~63%
• Proven using theories of submodular Proven using theories of submodular functions (diminishing returns)functions (diminishing returns)
• Applies to both diffusion modelsApplies to both diffusion models
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Testing on network dataTesting on network data
• Co-authorship networkCo-authorship network
• High-energy physics theory section High-energy physics theory section of of www.arxiv.org
• 10,748 nodes (authors) and ~53,000 10,748 nodes (authors) and ~53,000 edgesedges
• Multiple co-authored papers listed as Multiple co-authored papers listed as parallel edges (greater weight)parallel edges (greater weight)
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Testing on network dataTesting on network data
• Linear Threshold: influence weighed Linear Threshold: influence weighed by # of parallel lines, inversely by # of parallel lines, inversely weighed by degree of target node: w weighed by degree of target node: w = c= cu,v u,v /d/dvv
• Independent Cascade: Independent Cascade: pp set at 1% set at 1% and 10%; total probability for and 10%; total probability for u u vv is is
1 – (1 – 1 – (1 – pp)^c)^cu,vu,v
• Weighted Cascade: Weighted Cascade: pp = 1/ d = 1/ dvv
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AlgorithmsAlgorithms
• Greedy hill-climbingGreedy hill-climbing
• High degree: nodes with greatest High degree: nodes with greatest number of edgesnumber of edges
• Distance centrality: lowest average Distance centrality: lowest average distance with other nodesdistance with other nodes
• RandomRandom
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AlgorithmsAlgorithms
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Results: Linear Threshold Results: Linear Threshold ModelModel
Greedy: ~40% better than central, ~18% better than high degree
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Results: Weighted Cascade Results: Weighted Cascade ModelModel
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Results: Independent Results: Independent Cascade, Cascade, pp = 1% = 1%
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Results: Independent Results: Independent Cascade, Cascade, pp = 10% = 10%
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Advantages of Random Advantages of Random SelectionSelection
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Generalized modelsGeneralized models
• Generalized Linear Threshold: for node Generalized Linear Threshold: for node vv, influence of neighbors not necessarily , influence of neighbors not necessarily sum of individual influencessum of individual influences
• Generalized Independent Cascade: for Generalized Independent Cascade: for node node vv, probability , probability pp depends on set of depends on set of vv’s neighbors that have previously tried ’s neighbors that have previously tried to activate to activate vv
• Models computationally equivalent, Models computationally equivalent, impossible to guarantee approximationimpossible to guarantee approximation
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Non-Progressive Threshold Non-Progressive Threshold ModelModel• Active nodes can become inactiveActive nodes can become inactive
• Similar concept: at each time Similar concept: at each time tt, whether , whether or not or not vv becomes/stays active depends becomes/stays active depends on if influence meets thresholdon if influence meets threshold
• Can “intervene” at different times; need Can “intervene” at different times; need not perform all interventions at not perform all interventions at tt = 0 = 0
• Answer to progressive model with graph Answer to progressive model with graph G equivalent to non-progressive model G equivalent to non-progressive model with layered graph Gwith layered graph Gττ
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General Marketing General Marketing StrategiesStrategies• Can divide up total budget Can divide up total budget κκ into into
equal increments of size equal increments of size δδ
• For greedy hill-climbing strategy, can For greedy hill-climbing strategy, can guarantee performance within factor guarantee performance within factor of of
1 – e^[-(1 – e^[-(κκ **γγ)/()/(κκ ++ δδ **nn)])]
• As As δδ decreases relative to decreases relative to κκ, result , result approaches 1 – eapproaches 1 – e-1-1 = 63% = 63%
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Strengths of paperStrengths 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 datasetresults using real dataset
• Demonstrated that greedy hill-climbing Demonstrated that greedy hill-climbing strategy could guarantee results within strategy could guarantee results within 63% of optimum63% of optimum
• Used specific and generalized versions Used specific and generalized versions of two different diffusion modelsof two different diffusion models
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Weaknesses of paperWeaknesses of paper
• Doesn’t fully explain methodology of Doesn’t fully explain methodology of greedy hill-climbing strategygreedy hill-climbing strategy
• Lots of work not shown – simply refers Lots of work not shown – simply refers to work done in other papersto work done in other papers
• Threshold value uniformly distributed?Threshold value uniformly distributed?
• Influence inversely weighted by Influence inversely weighted by degree of target?degree of target?
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Questions?Questions?