Unformatted text preview: e procedure in which at each time k ≥ 0, the authority
and hub weights are updated according to:
bk +1 = (AT A)bk ,
hk +1 = (AAT )hk .
Using an eigenvector decomposition, one can show that this iteration
converges to a limit point related to the eigenvectors of the matrices AT A
and AAT .
Implementation of this algorithm requires “global knowledge,” therefore it is
implemented in a “querydependent manner”.
19 Networks: Lecture 7 Page Rank–1
Multibillion queryindependent idea of Google. Each node (or page) is important if it is cited by other important pages. Each node j has a single weight (PageRank value) w (j ) which is a function of the weights of his (incoming) neighbors: w (j ) = w (i ) ∑ dout (i ) Aij ,
i where A is the adjacency matrix, dout (i ) is the outdegree of node i (used
to dilute the importance of node i if he is linked to many nodes).
We can express this in vectormatrix notation as:
w T = w T P,
A where Pij = d ij(i ) (note that ∑j Pij = 1).
out
20 Networks: Lecture 7 Page Rank–2
This deﬁnes a random walk on the nodes of the network:
A walker chooses a starting node uniformly at random. Then, in each
step, the walker follows an outgoing link selected uniformly at random
from its current node, and it moves to the node that this link points to.
The PageRank of a node i is the limiting probability that a random walk on the network will end up at i as we run the walk for larger number of steps. Diﬃculty with PageRank: Dangling ends which may cause the random walk to get trapped. To ﬁx this, we allow the random walk at each step with probability (1 − s ) to jump to a node chosen uniformly at random (with probability s , as before, it follows a random outgoing link). This yields the following iteration for the PageRank algorithm: at step k , (1 − s ) T
e,
where e T = [1, . . . , 1].
n
This is the version of PageRank used in practice (with many other tricks)
with s usually between 0.8 and 0.9.
T
T
wk +1 = swk P + 21...
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 Fall '09
 Acemoglu
 Jon Kleinberg, PageRank, HITS algorithm, decentralized search

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