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Unformatted text preview: ility proportional to d (v , w )−α .
When α = 0, we have the uniform distribution over longrange contacts
– the distribution used in the model of WS.
As α increases, the longrange 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 longrange contacts and
randomly chosen starting and target nodes).
Given this setup, 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
longrange contact, and the location of the target (crucially, he does not know
the longrange 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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 Fall '09
 Acemoglu

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