04-search_annot

04-search_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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k so paths are short Ok, so paths are short And people are able to find them! ithout the global knowledge of the network) (without the global knowledge of the network) 9/29/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2
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nly knows locations of its friends s only knows locations of its friends and location of the target t oes not know links s does not know links of anyone but itself eographic navigation: rwards Geographic navigation: s forwards the message to the node closest to t 9/29/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 3
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Model: Grid where each node has one random edge This is a small world. Fact: A decentralized search algorithm in Watts Strogatz model needs n 2/3 steps to reach t in expectation p p Note: even though paths of O(log n) steps exist (it is really ~N 1/3 , where N=n 2 =|V| ) bout the proof: About the proof: Let’s do this for 1 dimensional case Setting: n nodes on a ring plus one random directed edge per node. Lower bound on search time is now ~ n 1/2 For d dimensional case: ~ n d/d+1 roof strategy rinciple of deferred decision: Proof strategy Principle of deferred decision: Doesn’t matter when a rand. decision is made if you haven’t seen it yet Assume random long range links are only created once you get to them 9/29/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 4
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Claim: /2 Expected search time is n 1/2 Let: E i = event that long link ut of node oints to some out of node i points to some node in interval I hen: = 2x/n Then: P (E i ) 2x/n (haven’t seen node i yet, but can assume random generation) t: = vent that any of first Let: E event that any of first k nodes you see has a link to I : Then: P(E) = P ( i k E i ) i k P(E i )=2kx/n 9/29/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 5
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(E)=P( kx/n P(E)=P( i k E i ) 2kx/n Need k,x s.t. 2kx/n < 1 hoose: x= n ½ Choose: k=x= ½n So, P(E) 2kx/n 2(½n n= =2(½n ½ )/n= ½ Suppose s is outside I and E does not happen Then the alg. must take min(k,x) steps to get to I 9/29/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 6
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Claim: Getting from s to t takes k=½n ½ steps If we don’t take a long link, we must raverse n ½ eps to get in . taese ½n steps to get Expected time to get to I: ½n 1/2 P(E doesn’t occur) = = n 1/2 ¼ Algorithm: Walk in the direction of t ith prob /n 1/2 e have a link to With prob. 1/n we have a link to I It takes n 1/2 steps on average to find such link fter that you use another /2 After that you use another n 1/2 steps to walk towards t 9/29/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 7
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04-search_annot - CS224W:SocialandInformationNetworkAnalysis

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