Four ways to compute the Google Pagerank

As described in THE $25,000,000,000 EIGENVECTOR THE LINEAR ALGEBRA BEHIND GOOGLE, we can compute the score of a page on a web as the maximal eigenvector of the matrix



where A is the scaled connectivity matrix of a web, S is an n × n matrix with all entries 1/n and m is a real number between 0 and 1.

Here's implemented four ways to compute the maximal eigenvector of the matrix using the numpy:

from numpy import *

def powerMethodBase(A,x0,iter):
 """ basic power method """
 for i in range(iter):
  x0 = dot(A,x0)
  x0 = x0/linalg.norm(x0,1)
 return x0

def powerMethod(A,x0,m,iter):
 """ power method modified to compute
     the maximal real eigenvector 
     of the matrix M built on top of the input matrix A """
 n = A.shape[1]
 delta = m*(array([1]*n,dtype='float64')/n) # array([1]*n is [1 1 ... 1] n times
 for i in range(iter):
  x0 = dot((1-m),dot(A,x0)) + delta
 return x0

def maximalEigenvector(A):
 """ using the eig function to compute eigenvectors """
 n = A.shape[1]
 w,v = linalg.eig(A)
 return abs(real(v[:n,0])/linalg.norm(v[:n,0],1))

def linearEquations(A,m):
 """ solving linear equations 
     of the system (I-(1-m)*A)*x = m*s """
 n = A.shape[1]
 C = eye(n,n)-dot((1-m),A)
 b = m*(array([1]*n,dtype='float64')/n)
 return linalg.solve(C,b)

def getTeleMatrix(A,m):
 """ return the matrix M
     of the web described by A """
 n = A.shape[1]
 S = ones((n,n))/n
 return (1-m)*A+m*S

A = array([ [0,     0,     0,     1, 0, 1],
            [1/2.0, 0,     0,     0, 0, 0],
            [0,     1/2.0, 0,     0, 0, 0],
            [0,     1/2.0, 1/3.0, 0, 0, 0],
            [0,     0,     1/3.0, 0, 0, 0],
            [1/2.0, 0,     1/3.0, 0, 1, 0 ] ])

n = A.shape[1] # A is n x n
m = 0.15
M = getTeleMatrix(A,m)

x0 = [1]*n
x1 = powerMethod(A,x0,m,130)
x2 = powerMethodBase(M,x0,130)
x3 = maximalEigenvector(M)
x4 = linearEquations(A,m)

# comparison of the four methods
labels = range(1,6)
print array([labels, x1, x2, x3, x4]).T

The matrix A used to the test the program describe the following web

The scores are (the first column show the labels):

[[ 1.          0.32954577  0.32954577  0.32954577  0.32954577]
 [ 2.          0.16505695  0.16505695  0.16505695  0.16505695]
 [ 3.          0.0951492   0.0951492   0.0951492   0.0951492 ]
 [ 4.          0.12210815  0.12210815  0.12210815  0.12210815]
 [ 5.          0.05195894  0.05195894  0.05195894  0.05195894]
 [ 6.          0.23618099  0.23618099  0.23618099  0.23618099]]

How to create a chart with Google Chart API

The example shows how to create a scatter plot using the Google Chart API.

import random
import urllib

def list2String(x):
 """ from a list like [1,2,5]
     return a string like '1,2,5' """
 data = ""
 for i in x:
  data += str(i)+","
 return data[0:len(data)-1]

def makeChart(x,y,filename):
 query_url = "http://chart.apis.google.com/chart?chxt=x,y&chs=300x200&cht=s&chd=t:"
 query_url += list2String(x)+"|"+list2String(y)
 chart = urllib.urlopen(query_url) # retrieve the chart
 print "saving",query_url
 f = open(filename,"wb")
 f.write(chart.read()) # save the pic
 f.close()

x = random.sample(range(0,100),10) # list with
y = random.sample(range(0,100),10) # random values in [0 100[
makeChart(x,y,"chart.png")

You can embed the picture in a web page:

<img alt="Google chart example" src="http://chart.apis.google.com/chart?chxt=x,y&amp;chs=300x200&amp;cht=s&amp;chd=t:64,10,18,42,49,83,73,27,44,51|77,89,13,87,27,34,38,44,22,42" />

Or use it from the disk.