Page rank algorithm

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Page Rank AlgorithmJung Hoon Kim

N5, Room 2239 E-mail: junghoon.kim@kaist.ac.kr

2014.01.14

Introduction

First introduced by Sergey Brin & Larry Page in 1998

Original ranking algorithm didn’t suitable for web in 1996# of Web pages grew rapidly

in 1996, query “classification technique” => 10 million relevant page searched!

content similarity method are easily spammed vulnerable for spam page

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Basic

page rank algorithm has two principleA hyperlink from a page pointing to another page is an

implicit conveyance of authority to the target page. thus, the more in-links that a page i receives, the more prestige the page i has

Pages that point to page i also have their own prestige score. A page with higher prestige score pointing to i is more important than a page with a lower prestige score pointing to i

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principle

hyperlink trick

many incident node means more important

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Authority

more authority people say .. is more important

John is computer scientistAlice is cooker

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Big picture

big picture

famous person is means having many incident edges

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Cyclic problem

In web, there are many cycles like this

this matrix has cycle A->B->Eit means the score is increased by infinitely

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Random suffer trick

To avoid many problem and many reasonthey adapted random surfer

each node can ability to move any node it can solve cycle problem high incident node can have high rank sometimes it called as damping factor(d)

by google initial model, d = 0.15

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Test

1000 times test resultnearly correct ;D, A has high rank

A has only one incident link

To easily identify rank, to express percentage is good methods

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Example

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Solve cycle problem

Solve cycle problem

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Formula

P(i) = Score of i page= Number of outlink of j i

a1

b3

c2

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Formula

in mathematically, we have a system of n linear equations.P=(P1, P2, P3 , … Pn)

A is adjacent matrix, so we can make this formula

Example

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Linear Algebra

formula

P is an eigenvector with the corresponding eigenvalue of 1. 1 is the largest eigenvalue and the PageRank vector P is the

principle eigenvector to calculate P, we can use power iteration algorithm

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Condition

but the conditions are that A is a stochastic matrix and that it is irreducible and aperiodic

We can see the graph model as markov modeleach web page is node and hyperlink is transition

A is not a stochastic matrix, because there are zero row(5). zero row means no out-link. So we fix the problem by adding a complete set of outgoing

links from each such page i to all the pages on the Web

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Modified version

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irreducible

if there is no path from u to v, A is not irreducible because of some pair of nodes u and v.if there are path u to v, A is irreducible!

A state i is periodic with period k > 1 if k is the smallest number such that all paths leading from state i back to state i have a length that is a multiple of k. If a state is not periodic, A markov chain is aperiodic if all states are aperiodic

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Page Rank

It is easy to deal with the above two problems with a single strategyWe add a link from each page to every page and give each

link a small transition probability controlled by a parameter d

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Page Rank

The computation of pagerank values of the Web pages can be done using the power iteration method, which produces the principal eigenvector with an eigenvalue of 1

The iteration ends when the PageRank values do not change much or converge.

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Real Page rank

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To deal with web spam is most important thinggive equal random surfer constants and calculate all the

page needs to many times to calculate itCurrently, Google use more 200 factors to calculate

ranking in web

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Thank you