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D d d 1 d 1 where the last relation follows by an

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Unformatted text preview: box. steps in expectation to find a node with a long-range contact in 11 Networks: Lecture 7 Proof Idea for α = 2 We again first compute the proportionality constant Z : n n 1 1 Z = ∑ d 2 = ∑ ≈ log(n ). d d d =1 d =1 (where the last relation follows by an integral approximation). Instead of picking an “impenetrable box” around the target, we show that it is easy to enter smaller and smaller sets centered around the target node. We consider a node at a distance 2s from the target and compute the time it takes to halve this distance (i.e., get into a box of size s ). 12 Networks: Lecture 7 Proof Idea for α = 2 (Continued) The probability of having a long range contact in the box of size s is ≥ s 2 log1 n) (21)2 ≈ log1 n) . ( ( s Hence, it takes in expectation log(n ) time steps to halve the distance. Since we can halve the distance at most log(n ) times, this leads to an algorithm that takes (log(n ))2 steps to reach the target in expectation. 13 Networks: Lecture 7 Proof Idea for 2 < α < 3 Computing the proportionality constant Z in this case yields Z = constant (independent of n) for large n. The expected length of a typical long-range contact is given by E[length of a typical long range contact] = 1n 1 ∑ d .d . d α ≈ n3−α . Z d =1 Hence, the expected time is at least n /n3−α = nα−2 . 14 Networks: Lecture 7 Web Search – Link Analysis When you go to Google and type “MIT”, the first result it displays is web.mit.edu, the home page of MIT University. How does Google know that this was the best answer? This is a problem of information retrieval: since 1960’s, automated information retrieval systems were designed to search data repositories in response to keyword queries. Classical approach has been based on “textual analysis”, i.e., look at each page separately without regard to the link structure. In the example of MIT home page, what makes it stand out is the number of links that “point to it”, which can be used to assess...
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