cloud computing project

Page Rank Implementation

CLOUD COMPUTING PROJECT

-Team 3

By: - Devendra Singh Parmar

Project Abstract

Instructor: Prof. Reddy RajaMentor: Ms M.Padmini

To Implement PageRank Algorithm using Map-Reduce for Wikipedia and verify it for smaller data-sets

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

Motivation

-> Need for PageRank:

The Search engines store billions of web pages which overall contain trillions of web url links. So, there is a need for an algorithm that gives the most relevant pages specific to a query.

-> Need for Distributed Environment( Map-Reduce and Distributed Storage)

• Trillions of links implies huge data storage required. (if each url requires 0.5K, then we need over 400TB just to store URLs!) • Large data set implies large computations

Thus, we handle above issues in our project by using a distributed cluster

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

Introduction

PageRank is a link analysis algorithm, named after Larry Page, used by the Google Internet search engine that assigns a numerical weighting to each element of a hyperlinked set of documents, such as the Worldwide Web, with the purpose of "measuring" its relative importance within the set

The numerical weight that it assigns to any given element E is also called the PageRank of E and denoted by PR(E).

Algorithm

Google figures that when one page links to another page, it is effectively casting a vote for the other page. The more votes that are cast for a page, the more important the page must be. Also, the importance of the page that is casting the vote determines how important the vote itself is. Google calculates a page's importance from the votes cast for it. How important each vote is also taken into account when a page's PageRank is calculated.

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

The PageRank Equation

Simple Iterative Algorithm

For kth iteration PageRank of ith page is given by:

Here,

The PageRank Equation(Issues and Enhancement)

Problems:

• Rank Sinks or Dangling Pages• Cycles

Solution:

PageRank Equation(Enhancement)

Solution for Cycles and If a random surfer gets bored

Here ‘d ‘ is known as damping factor . It represents the probability, at any step, that the person will continue surfing . The value of ‘d’ is typically kept 0.85

PageRank Equation (finally)

In other words

In a simpler way:- a page's PageRank = 0.15 /N+ 0.85 * (a "share" of the PageRank of every page that links to it) "share" = the linking page's PageRank divided by the number of outbound links on the page. And N= the number of documents in collection

The equation of PageRank shows clearly how a page's PageRank is arrived at. But what isn't immediately obvious is that it can't work if the calculation is done just once.

PageRank Equation-as per the published paper :“The Anatomy of a Large-Scale Hyper textual Web Search Engine”-Sergey Brin and Lawrence Page

We assume page A has pages T1...Tn which point to it (i.e., are citations). The parameter d is a damping factor which can be set between 0 and 1. We usually set d to 0.85.. Also C(A) is defined as the number of links going out of page A. The PageRank of a page A is given as follows:

PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))

->Note that the PageRanks form a probability distribution over web pages, so the sum of all web pages’ PageRanks will be one.

IssuesIn the Original Formula

Formula given in the in Page and Brin's paper does not supports the statement that "the sum of all PageRanks is one“

Hence to support the statement the formula is modified as:

PR(A) = (1-d)/N + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))

where N= the number of documents in collection

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

Brief Description of Project

Input: Data Set containing multiple records where each record contains the Url of the Page(from Url) followed by the url of a page to which it is pointing to(ToUrl).

FromUrl

Wiki_Votes.txt

ToUrl

Brief Description of Project(Contd.)

Output:The output file consist of records containing the url of the page(from Url), the page rank value of the page(PRValue) and the list of urls to which the page points to(ToUrlList).

FinalOutput.txt

fromUrl ToUrlListPRValue

Brief Description of ProjectModules

Module1: Converter

Module2: PageRank Calculator

Module3: Output Analyzer

WebGraph

Converter

PageRank

Calculator

Iterateuntil convergence

Output Analyzer

Search Engine

...

CreateIndex

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

Module1: ConverterInput-Output

Converter (Initializing with PR= 1/N )FromUrl PRValue

List:

Module1: ConverterIssues

Self Loops:

-handled by checking the FromUrl with ToUrl before sending it to the reduce function

Dangling Pages:

-handled by initializing their PRValue with 1/N and the List of ToUrls is left blank.

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

Module2: PageRank CalculatorInput-Output

PageRank Calculator (User can give Precision)

Module2: PageRank CalculatorMap: Input: index.html PRValue OutList: <

1.html 2.html... > Output 1. Output for each outlink: key: “1.html” value: PRValue/ ListLength (Vote Share) 2. ToUrl itself key: index.html value: <OutList>

Reduce Input:

Key: “1.html” Value: 0.5 23

Value: 0.24 2……. Value : UrlList <OutLink>

Output: Key: “1.html” Value: “<new pagerank> <OutList>

1.html 2.html...”

Start with the initial PageRank and Outlinks of a document.

n

i i

i

tC

tPRd

N

dxPR

1 )(

)()1()(

Module2: PageRank CalculatorMap: Input: index.html PRValue OutList: <

1.html 2.html... > Output 1. Output for each outlink: key: “1.html” value: PRValue/ ListLength (Vote Share) 2. ToUrl itself key: index.html value: <OutList>

Reduce Input:

Key: “1.html” Value: 0.5 23

Value: 0.24 2……. Value : UrlList <OutLink>

Output: Key: “1.html” Value: “<new pagerank> <OutList>

1.html 2.html...”

n

i i

i

tC

tPRd

N

dxPR

1 )(

)()1()(

For each Outlink, output the PageRank’s share of the Inlinks, and List of outlinks.

Module2: PageRank CalculatorMap: Input: index.html PRValue OutList: <

1.html 2.html... > Output 1. Output for each outlink: key: “1.html” value: PRValue/ ListLength (Vote Share) 2. ToUrl itself key: index.html value: <OutList>

Reduce Input:

Key: “1.html” Value: 0.5 23

Value: 0.24 2……. Value : UrlList <OutLink>

Output: Key: “1.html” Value: “<new pagerank> <OutList>

1.html 2.html...”

n

i i

i

tC

tPRd

N

dxPR

1 )(

)()1()(

Now the reducer has a Url of document, all the inlinks to that document and their corresponding PageRank’s share and List of outlinks.

Module2: PageRank CalculatorMap: Input: index.html PRValue OutList: <

1.html 2.html... > Output 1. Output for each outlink: key: “1.html” value: PRValue/ ListLength (Vote Share) 2. ToUrl itself key: index.html value: <OutList>

Reduce Input:

Key: “1.html” Value: 0.5 23

Value: 0.24 2……. Value : UrlList <OutLink>

Output: Key: “1.html” Value: “<new pagerank> <OutList>

1.html 2.html...”

n

i i

i

tC

tPRd

N

dxPR

1 )(

)()1()(

Compute the new PageRank and output in the same format as the input.

Module2: PageRank CalculatorMap: Input: index.html PRValue OutList: <

1.html 2.html... > Output 1. Output for each outlink: key: “1.html” value: PRValue/ ListLength (Vote Share) 2. ToUrl itself key: index.html value: <OutList>

Reduce Input:

Key: “1.html” Value: 0.5 23

Value: 0.24 2……. Value : UrlList <OutLink>

Output: Key: “1.html” Value: “<new pagerank> <OutList>

1.html 2.html...”

n

i i

i

tC

tPRd

N

dxPR

1 )(

)()1()(Now iterate until

convergence (determined by the precision value).

Module2: PageRank Calculator IssuesCatch22 Situation

Suppose we have 2 pages, A and B, which link to each other, and neither have any other links of any kind. This is what happens:-

Step 1: Calculate page A's PageRank from the value of its inbound links

Step 2: Calculate page B's PageRank from the value of its inbound links

we can't work out A's PageRank until we know B's PageRank, and we can't work out B's PageRank until we know A's PageRank. Thus the PageRank of A and B will be inaccurate.

Module2: PageRank Calculator IssuesCatch22 situation (solution)

This problem is overcome by repeating the calculations many times. Each time produces slightly more accurate values. In fact, total accuracy can never be achieved because the calculations are always based on inaccurate values.The number of iterations should be sufficient to reach a point where any further iterations wouldn't produce enough of a change to the values to matter.

=> Use “delta function” which will keep track of changes in the PageRank of all the pages and if the change in PageRank of all the pages is less than the value specified by the user the iterations can be stopped.

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications

Module 3: Output AnalyzerInput-Output

Input

Analyzer ( If user want Top 3)

Output

Agenda

Motivation Introduction to Algorithm PageRank Equation Analysis Brief Description of Project Module1 Module2 Module3 Applications Questions

Applications and ExtensionsA simple model of Search Engine. (Implemented)

The application utilizes: 1. The PageRank calculated by the PageRank

Calculator2. The output generated by a map-reduce

module that finds out the number of times a pattern (as per the user’s query) matches in each of the files present in data set.

And outputs:The list of pages which are relevant to the query made in the order of their importance.

(DEMO)

Applications and ExtensionsOther Applications:

• PageRank-based mechanism to rank knowledge items used in E-Learning.

• GeneRank (based on PageRank) ranks the genes analyzed in the microarray to see the relationship between the cell’s function and gene expression.

• Can be used to sort the items present in the side menu in various blogs and sites depending on their importance.

References

http://infolab.stanford.edu/pub/papers/google.pdf ( research paper by Brin and Page)

http://www.ams.org/featurecolumn/archive/pagerank.html

http://en.wikipedia.org/wiki/PageRank

http://www.webworkshop.net/pagerank.html#how_is_pagerank_calculated

Questions

Thank You