PageRank

1
PageRank
The Idea

• Create a device dissection lab that dissects a
  real world mathematical object
• Shows students uses of Mathematics in the world
• Hands on experiments.
• Helps peak students interest in Mathematics.

Ian Stickles imstickl_at_edisto.cofc.edu or
imstickles_at_yahoo.com Major Pure
Mathamatics Professor Dr. Langville
Stochastic Matrix H 1/n e aT
Hyperlink Matrix
0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0
0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0
0 0 1 0
Google Matrix alpha S (1-alpha) 1/n e
eT
.015 .865 .015 .015 .015 .015 .015 .015 .015 .015
.015 .015 .865 .015 .015 .015 .015 .015 .015 .015
.865 .015 .015 .015 .015 .015 .015 .015 .015 .01
5 .015 .015 .44 .015 .015 .44 .015 .015 .015 .015
.015 .015 .015 .015 .015 .44 .44 .015 .015 .015
.015 .015 .015 .44 .015 .015 .015 .015 .015 .44
.015 .015 .015 .015 .015 .015 .015 .44 .44 .015 .
1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .015 .015 .015 .015
.015 .44 .015 .44 .015 .015 .015 .015 .015 .015 .
44 .015 .015 .015 .44 .015
PageRank vector vk vT(k-1) G
.43 .015 .045 .2 .15 .025 .02 .1
.0075 .0075
This Module

• PageRank Model.
• Introduces Markov chains.
• Explains PageRank.
• Exercises used to effects of changing formulas.
• Uses MatLab.
• Further readings.

PageRank

• So how is PageRank determined?
• First, the google matrix has to be calculated.
  This allows each page to be reached by every
  other page.
• Second, an iterative method must be used to
  calculate the rank.
• The power method is used by giving the each page
  an initial page rank of 1/n (This is considered a
  fair ranking).
• This method is iterated a specific number of
  times to insure that the PageRank vector has
  converged.
• Once the PageRank vector has converged it must
  undergo another alteration that will give it a
  rank between 0 and 10.
• A content score is also added to the rank of each
  page to help calculate the rank. This helps when
  scores are relatively close.

PageRank, not just for the Web?

• The possible uses of the PageRank Algorithm reach
  far beyond just that of ranking web pages
• Possible rankings in which PageRank could be used
• Ranking web pages
• Ranking sports teams
• Ranking cars
• Ranking classes
• Ranking Professors? (all 10s Im sure.)

PageRank Calculations, Another Look

• Calculating the Google Matrix is time expensive
  and cost a lot in computer resources
• Google doesnt actually calculate the Google
  Matrix or the Stochastic Matrix
• Instead Google uses the row-normalized Hyperlink
  Matrix and performs all its calculations on
  that.
• The PageRank vector is also independent of the
  starting PageRank vector (The stationary vector).
• The formula
• AlphaH (alphaa (1-alpha)e)1/neT