10-celf_annot

10-celf_annot - CS224W:SocialandInformationNetworkAnalysis

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CS224W: Social and Information Network Analysis re eskovec tanford University Jure Leskovec, Stanford University http://cs224w.stanford.edu
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ost influential set of 0.4 Most influential set of size k: set S of k nodes roducing rgest b a d 0.4 0.2 0.2 0.3 0.3 0.3 producing largest expected cascade size S) activated e f h 0.4 0.2 0.4 0.3 0.3 0.3 0.2 f(S) if activated [Domingos Richardson ‘01] c g i 0.4 Optimization problem: ) ( max S f 10/20/2010 2 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu k size of S
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Hill Climbing: f(S i 1 {v}) Start with S 0 ={} For i=1…k Choose node v that max f(S {v}) a b i 1 Let S i = S i 1 {v} Hill climbing produces a solution S where S) / ) f ti l l (~63%) c d f(S) (1 1/e) of optimal value (~63%) [Hemhauser, Fisher, Wolsey ’78, Kempe, Kleinberg, Tardos ‘03] Claim holds for functions f with 2 properties: e f is monotone: if S T then f (S) f (T) and f ({})=0 bmodular f is submodular: adding element to a set gives less improvement than adding to one of subsets 10/20/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 3
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ext we must show bmodular Next, we must show f is submodular : S T S }) S) f(T }) T) Gain of adding a node to a small set Gain of adding a node to a large set f(S {u}) – f(S) {u}) – f(T) Basic fact: If f ,…,f are submodular , and c ,…,c 0 1 K 1 k then i c i f i is also submodular 4 10/20/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S }) S) f(T }) T) S T : simple bmodular nction: Gain of adding u to a small set Gain of adding u to a large set f(S {u}) – f(S) {u}) – f(T) A simple submodular function: Sets S 1 ,…,S m S) | i f i f t S f(S) = | i S S i | (size of union of sets S i , i S) T S u 10/20/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 5
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rinciple of deferred a d Principle of deferred decision: Generate randomness b e f ahead of time ip a coin for each c g h i Flip a coin for each edge to decide whether it ill succeed when (if ever) it attempts to transmit will succeed when (if ever) it attempts to transmit Edges on which activation will succeed are live ) = size of the set reachable by f i (S) size of the set reachable by live edge paths from nodes in S (e.g., ={a,d}) 10/20/2010 6 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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x outcome f coin flips a d Fix outcome i of coin flips f i (S) = size of cascade from S given b e f these coin flips f i (v) = set of nodes c g h i reachable from v on live edge paths f i (S) = | v S f i (v) | f i (S) is submodular xpected influence set size: Expected influence set size: f (S)= i f i (S) f is submodular!
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This note was uploaded on 01/11/2011 for the course CS 224 at Stanford.

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10-celf_annot - CS224W:SocialandInformationNetworkAnalysis

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