Page rank algorithm

1KAIST Knowledge Service Engineering

Data Mining Lab.

Page Rank AlgorithmJung Hoon Kim

N5, Room 2239 E-mail: [email protected]

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


Data Mining Lab.

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


Data Mining Lab.

principle

hyperlink trick

many incident node means more important


Data Mining Lab.

Authority

more authority people say .. is more important

John is computer scientistAlice is cooker


Data Mining Lab.

Big picture

big picture

famous person is means having many incident edges


Data Mining Lab.

Cyclic problem

In web, there are many cycles like this

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


Data Mining Lab.

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


Data Mining Lab.

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


Data Mining Lab.

Example


Data Mining Lab.

Solve cycle problem

Solve cycle problem


Data Mining Lab.


Data Mining Lab.

Formula

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

a1

b3

c2


Data Mining Lab.

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


Data Mining Lab.

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


Data Mining Lab.

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


Data Mining Lab.

Modified version


Data Mining Lab.

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


Data Mining Lab.

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


Data Mining Lab.

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.


Data Mining Lab.

Real Page rank


Data Mining Lab.

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


Data Mining Lab.

Thank you

Technology

Page rank algorithm