Page Rank

Overview• Two dimensional arrays• Monte Carlo algorithms• Searching the world wide web• Big data• Page rankGoal: we will write a program to compute the relevancy of WWW documents based on the static structure of the WWW.

Two Dimensional Arrays• Significance (a topic on the AP Computer Science A exam)• Syntax• Example of matrix multiplication• Arrays of arrays

Significance of Two Dimensional Arrays• Tables; for instance, assignments for each student in a class, quarterly

sales for each item in inventory, etc.• Matrices and binary relations in mathematics. For example, is there a

direct road from city1 in USA to city2 in USA?• For our goal in the this section, we will have need for the number of

links from doc1 in the WWW to doc2 in the WWW.

Syntax• int[][] frequency = new int [26][26];• Elements are accessed: frequency[4][7] and not frequency[4,7]• Array indices in Java (like C, C++, C#) always begin with 0; in other

words, the element with index 1 is the second element of the array.

Matrix multiplication

Matrix Multiplication Exercise• http://cs.fit.edu

/~ryan/java/programs/basic_algorithms/MatrixMultiplication2.java

Arrays of Arrays• Logically: arrays of arrays in the tradition of C and C++. Very simple.• Unfortunately: introduces pointers, memory allocation, etc. Very

complicated.

Monte Carlo Methods• Introduction• The example of a Monte Carlo estimate for Pi (Java exercise). Fair

shuffling (Java exercise). Random walk (important in financial analysis)• Used in path tracing to create realistic images• Percolation – an example of the power of a Monte Carlo algorithmGoal: we will write a Monte Carlo algorithm to estimate the relevancy of WWW documents based on the static structure of the WWW.

Monte Carlo Casino• The name refers to the grand casino in the

Principality of Monaco at Monte Carlo, which is well-known around the world as an icon of gambling.

Monte Carlo estimate for PiJava exercise: http://cs.fit.edu/~ryan/java/programs/basic_algorithms/ComputePi2.java

Since we know the value of pi it is not really necessary to invent an algorithm to estimate its value.

Fair shuffling (Java exercise)

• How would you test a algorithm for shuffling, say, cards? In particular how would you know if all of the many possible results are equally likely?• Main program

http://cs.fit.edu/~ryan/java/programs/basic_algorithms/Experiment.java. Nothing to write; requires the method to shuffle.• http://cs.fit.edu/~ryan/java/programs/basic_algorithms/Shuffle.java

contains two methods of shuffling cards.• Run the experiment with multiple trials and convince yourself both

methods are fair

Percolation TheoryPercolation. Pour liquid on top of some porous material.Will liquid reach the bottom? Many applications in chemistry, materials science, etc.• Spread of forest fires.• Natural gas through semi-porous rock.• Flow of electricity through network of resistors.• Permeation of gas in coal mine through a gas mask filter.

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Percolation TheoryGiven an N-by-N system where each site is vacant with probability p, what is the probability that system percolates?

Remark. Famous open question in statistical physics. No known mathematical solution. Computational thinking creates new science.Recourse. Take a computational approach: Monte Carlo simulation.Uses a recursive, dfs algorithm, but diverges from the present topic. (Recursion is a topic on the AP Computer Science A exam.)

p = 0.3(does not percolate)

p = 0.4(does not percolate)

p = 0.5(does not percolate)

p = 0.6(percolates)

p = 0.7(percolates)

We will examine a Monte Carlo algorithm for estimating the relevancy of WWW documents.

Random Walk• Page rank can be computed a lot like random walk• See the Java applet (1 dim) at

http://www.math.uah.edu/stat/applets/RandomWalkExperiment.html• See the Java applet (2 dim) at

http://vlab.infotech.monash.edu.au/simulations/swarms/random-walk/

Searching the World Wide Web• History of Search Engines• Hypertext• Crawling the World Wide Web• Indexing

History of Search Engines• History of Search by Larry Kim of WordStream

Markup and Hypertext• Documents served up through the WWW are generally “marked up”

for presentation in a structured, standard called hypertext markup language (HTML).• The most important feature of HTML is the referencing (via URLs) of

other WWW documents which enables easy, non-sequential, and varied paths of reading the documents.

Hypertext

WWW Spiders• Google, and others, continually, crawl around the WWW recording

what they see to enable searching.

44% of hits and 35% of bandwidth is attributable to bots (and other odd things).

July 2013 (up to 9:30 am 26 Jul 2013) on the WWW server cs.fit.edu

Russian search engine

Indexing• Finding a relevant document in a vast ocean of linked HTML

documents requires a very large index.• An index is a (sorted) list of keywords (terms) and the list of values

(URLs) which contain them.

An example index of WWW documentsBourgeois .../manifesto.txtHero …/lilwomen.txt, …/muchado.txt, …/war+peace.txtHis .../manifesto.txt, …/lilwomen.txt, …/mobydick.txt, …/muchado.txt, …/war+peace.txtTreachery …/war+peace.txtWhale …/mobydick.txtYellowish …/lilwomen.txt , …/war+peace.txt

Several Issues• Pick out the words from the mark-up• What’s a word? 2nd, abc’s, CSTA• Normalize: lowercase, stemming• Some words are not worth indexing• “the”, “a”, etc.• A so-called stop list, eg., words ignored in Wikipedia search• Java exercise: http://cs.fit.edu/~ryan/java/programs/xml/URLtoText.java

First some preliminary remarks before doing the exercise.

Searching and SortingProblem: Determine if the word is in the stop list. What is the best approach?

• Searching: linear search, binary search. (These are topics on the AP Computer Science A exam.) Binary search requires the data (the index, for example) to be sorted.• Sorting: selection sort, insertion sort, merge sort, quick sort; external

sorting. (The first three of these sorts are topics on the AP Computer Science A exam.)

Linear versus Binary search Suppose each

comparison takes one

millisecond (0.001)

Linear versus Binary Search

Linear versus Binary Search

Obama at Google• https://www.youtube.com/watch?v=k4RRi_ntQc8

Sorting Demo• http://cs.fit.edu/~ryan/cse1002/sort.html• See also sorting illustrated by Algo-rythmics

http://algo-rythmics.ms.sapientia.ro and folk dancers

Now do the exercise• Java exercise: http://cs.fit.edu/~ryan/java/programs/xml/URLtoText.java

• PS. How to students really program?• http://xkcd.com/1185 Observe the tool tip!

OK, we have a keyword index. It is likely we still have “gazillion” documents, for most of the terms. (See Googlewacks, Googlewhackblatt; one and two words search terms that return one document.)How do we find the most relevant pages?

Big Data• The problem• Count-Min Algorithm

The problem with Big DataConsider a popular website which wants to keep track of statistics on the queries used to search the site. One could keep track of the full log of queries, and answer exactly the frequency of any search query at the site. However, the log can quickly become very large. This problem is an instance of the count tracking problem. Even known sophisticated solutions for fast querying such as a tree-structure or hash table to count up the multiple occurrences of the same query, can prove to be slow and wasteful of resources. Notice that in this scenario, we can tolerate a little imprecision. In general, we are interested only in the queries that are asked frequently. So it is acceptable if there is some fuzziness in the counts. Thus, we can tradeoff some precision in the answers for a more efficient and lightweight solution. This tradeoff is at the heart of sketches.Cormode and Muthurishnon, 2011

Page Rank• Not finding pages, but ordering the found pages• Makes a big difference in the user’s experience, if “good” or “relevant”

pages come first.• The upcoming algorithm gave Google a competitive advantage• How would you rank pages? • The approach/algorithm called “page rank” is not based on the WWW

surfer as voter (popularity), but on the WWW author as voter (hence relatively static)• Conceptually in the Page Rank algorithm a random surfer mindlessly

follows the hyperlinks of the entire WWW

Input/Output• What is the input? The entire WWW modeled as a graph.• What is the output? The ranking of every page in the WWW.• By assigning one number to every page, then the search query will

order the found pages by the rankings in order to present to the user the most relevant pages first.

S&W Tiny Hypertext

S&W Tiny Graph

S&W Tiny: Adj list & Adj matrix50 11 2 1 21 3 1 3 1 42 33 04 0 4 2

5 5 0 1 0 0 0 0 0 2 2 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

PAGE RANK TINY

Wiki2 Hypertext

Wiki2 Graph

Wiki2: Adj List & Adj Matrix70 1 0 2 0 3 0 4 0 61 02 0 2 13 1 3 2 3 44 0 4 2 4 3 4 55 0 5 46 4

7 70 1 1 1 1 0 11 0 0 0 0 0 01 1 0 0 0 0 00 1 1 0 1 0 01 0 1 1 0 1 0 1 0 0 0 1 0 00 0 0 0 1 0 0

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

PAGE RANK WIKI2

Wiki1 Hypertext

Wiki1 Graph

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

PAGERANK WIKI1

Java Exercise• Modify Adajency1.java

1. Print adjacency matrix2. Print probability matrix3. Print probability matrix with 90-10 rule

Interactive WWW Page for PageRank• http://williamcotton.com/pagerank-explained-with-javascript

Reachability, Markov Theory

Can node 2 reach node 4? Yes, using a path of length 2 through node 3.

Final Challenge• Raise the page rank of page “23” by modifying only the links on page

“23”• Decrease the page rank of page “23” by modifying only the links on

page “23”• Can you find the maximum/minimum page rank?

Search engine optimization, link schemes, link farming, Google bombs

Ted Talks: Brin & Page: The Genesis of Google• http://www.ted.com/talks/

sergey_brin_and_larry_page_on_google.html