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When 0 we have the uniform distribution over long

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Unformatted text preview: ility proportional to d (v , w )−α . When α = 0, we have the uniform distribution over long-range contacts – the distribution used in the model of WS. As α increases, the long-range contact of a node becomes more clustered in its vicinity on the grid. 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: (left) Small clustering exponent, (right) large clustering exponent 7 Networks: Lecture 7 Decentralized Search in this Model We evaluate different search procedures according to their expected delivery time– the expected number of steps required to reach the target (where the expectation is over a randomly generated set of long-range contacts and randomly chosen starting and target nodes). Given this set-up, we will prove that decentralized search in WS model will necessarily require a large number of steps to reach the target (much larger than the true length of the shortest path). As a model, WS network is effective in capturing clustering and existence of short paths, but not the ability of people to actually find those paths. The problem here is that the weak ties that make the world small are “too random” in this model. The parameter α captures a tradeoff between how uniform the random links are. Question: Is there an optimal α (or network structure) that allows for rapid decentralized search? 8 Networks: Lecture 7 Efficiency of Decentralized Search Theorem (Kleinberg 2000) Assume that each node only knows his set of local contacts, the location of his long-range contact, and the location of the target (crucially, he does not know the long-range contacts of the subsequent nodes). Then: (a) For 0 ≤ α < 2, the expected delivery time of any decentralized algorithm is at least βα n(2−α)/3 for some constant βα . (b) For α = 2, there is a decentralized algorithm so that the expected delivery time is at most β2 (log (n ))2 for some constant β2 . (c) For 2 < α &l...
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