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If everybody knows the global network structure or if

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Unformatted text preview: f everybody knows the global network structure or if we can “flood the network” (i.e., everyone will send the letter to all their friends), we would be able to find the short paths efficiently. With local information, even if the social network has short paths, it is not clear that such decentralized search will be able to find them efficiently. a p b c o n d m e l f g j i h Image by MIT OpenCourseWare. Adapted from Easley, David, and Jon Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. New York, NY: Cambridge University Press, 2010. ISBN: 9780521195331. Figure: In myopic search, the current message-holder chooses the contact that is closest to the target and forwards the message to it. 5 Networks: Lecture 7 A Model for Decentralized Search—1 Kleinberg introduces a simple framework that encapsulates the paradigm of WS – rich in local connections with a few long range links. The starting point is an n × n two-dimensional grid with directed edges (instead of an undirected ring). The nodes are identified with the lattice points, i.e., a node v is identified with the lattice point (i , j ) with i , j ∈ {1, . . . , n }. For any two nodes v and w , we define the distance between them d (v , w ) as the number of grid steps between them, d ((i , j ), (k , l )) = |k − i | + |l − j |. Each node is connected to its 4 local neighbors directly – his local contacts. Each node also has a random edge to another node – his long range contact. Image by MIT OpenCourseWare. Adapted from Easley, David, and Jon Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. New York, NY: Cambridge University Press, 2010. ISBN: 9780521195331. Figure: A grid network with n = 6 and the contacts of a node u . 6 Networks: Lecture 7 A Model for Decentralized Search—2 The model has a parameter that controls the “scales spanned by the long-range weak ties.” The random edge is generated in a way that decays with distance, controlled by a clustering exponent α: In generating a random edge out of v , we have this edge link to w with probab...
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