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Unformatted text preview: box. steps in expectation to ﬁnd a node with a longrange 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 longrange 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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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.
 Fall '09
 Acemoglu

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