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from numpy import *
def transposeLinkMatrix(
outGoingLinks = [[]]
Transpose the link matrix. The link matrix contains the pages each page points to.
However, what we want is to know which pages point to a given page. However, we
will still want to know how many links each page contains (so store that in a separate array),
as well as which pages contain no links at all (leaf nodes).
@param outGoingLinks outGoingLinks[ii] contains the indices of pages pointed to by page ii
@return a tuple of (incomingLinks, numOutGoingLinks, leafNodes)
nPages = len(outGoingLinks)
# incomingLinks[ii] will contain the indices jj of the pages linking to page ii
incomingLinks = [[] for ii in range(nPages)]
# the number of links in each page
numLinks = zeros(nPages, int32)
# the indices of the leaf nodes
leafNodes = []
for ii in range(nPages):
if len(outGoingLinks[ii]) == 0:
numLinks[ii] = len(outGoingLinks[ii])
# transpose the link matrix
for jj in outGoingLinks[ii]:
incomingLinks = [array(ii) for ii in incomingLinks]
numLinks = array(numLinks)
leafNodes = array(leafNodes)
def pageRankGenerator(
At = [array((), int32)],
numLinks = array((), int32),
ln = array((), int32),
alpha = 0.85,
convergence = 0.01,
checkSteps = 10
Compute an approximate page rank vector of N pages to within some convergence factor.
@param At a sparse square matrix with N rows. At[ii] contains the indices of pages jj linking to ii.
@param numLinks iNumLinks[ii] is the number of links going out from ii.
@param ln contains the indices of pages without links
@param alpha a value between 0 and 1. Determines the relative importance of "stochastic" links.
@param convergence a relative convergence criterion. smaller means better, but more expensive.
@param checkSteps check for convergence after so many steps
# the number of "pages"
N = len(At)
# the number of "pages without links"
M = ln.shape[0]
# initialize: single-precision should be good enough
iNew = ones((N,), float32) / N
iOld = ones((N,), float32) / N
done = False
while not done:
# normalize every now and then for numerical stability
iNew /= sum(iNew)
for step in range(checkSteps):
# swap arrays
iOld, iNew = iNew, iOld
# an element in the 1 x I vector.
# all elements are identical.
oneIv = (1 - alpha) * sum(iOld) / N
# an element of the A x I vector.
# all elements are identical.
oneAv = 0.0
if M > 0:
oneAv = alpha * sum(iOld.take(ln, axis = 0)) / N
# the elements of the H x I multiplication
ii = 0
while ii < N:
page = At[ii]
h = 0
if page.shape[0]:
h = alpha * dot(
iOld.take(page, axis = 0),
1. / numLinks.take(page, axis = 0)
iNew[ii] = h + oneAv + oneIv
ii += 1
diff = iNew - iOld
done = (sqrt(dot(diff, diff)) / N < convergence)
yield iNew
def pageRank(
linkMatrix = [[]],
alpha = 0.85,
convergence = 0.01,
checkSteps = 10
Convenience wrap for the link matrix transpose and the generator.
incomingLinks, numLinks, leafNodes = transposeLinkMatrix(linkMatrix)
for gr in pageRankGenerator(incomingLinks, numLinks, leafNodes,
alpha = alpha,
convergence = convergence,
checkSteps = checkSteps):
final = gr
return final
#!/usr/bin/env python
from pageRank import pageRank
# a graph with one link from page one to page two
links = [
print pageRank(links, alpha = 1.0)
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