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Search Algorithms Winter Semester 2004/2005 22 Nov 2004 6th Lecture. Christian Schindelhauer [email protected]. Chapter III. Chapter III Searching the Web 22 Nov 2004. Searching the Web. Introduction The Anatomy of a Search Engine Google’s Pagerank algorithm The Simple Algorithm - PowerPoint PPT Presentation
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1
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Search AlgorithmsWinter Semester 2004/2005
22 Nov 20046th Lecture
Christian Schindelhauer
Search Algorithms, WS 2004/05 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Chapter III
Chapter IIISearching the Web
22 Nov 2004
Search Algorithms, WS 2004/05 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Searching the Web
Introduction
The Anatomy of a Search Engine
Google’s Pagerank algorithm
– The Simple Algorithm
– Periodicity and convergence
Kleinberg’s HITS algorithm
– The algorithm
– Convergence
The Structure of the Web
– Pareto distributions
– Search in Pareto-distributed graphs
Search Algorithms, WS 2004/05 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Overview Search Engineshttp://www.searchengineshowdown.com/(March 2002)
Number of documents
Search EngineShowdown
Estimate (millions)
Claim (millions)
Google 968 1,500
WiseNut 579 1,500
AllTheWeb 580 507
Northern Light 417 358
AltaVista 397 500
Hotbot 332 500
MSN Search 292 500
Search Algorithms, WS 2004/05 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Overview Search Engineshttp://www.searchengineshowdown.com/(Dez. 2002)
Number of documents
Search EngineShowdown
Estimate (millions)
Claim (millions)
Google 3,033 3,083
AlltheWeb 2,106 2,116
AltaVista 1,689 1,000
WiseNut 1,453 1,500
Hotbot 1,147 3,000
MSN Search 1,018 3,000
Teoma 1,015 500
NLResearch 733 125
Gigablast 275 150
Search Algorithms, WS 2004/05 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Problems of Searching the Web
Currently (Nov 2004) more than 8 billion = 8.000 millions web-pages– 10.000 words cover more than 95% of each text– much more web-pages than words– Users hardly ever look through more than 40 results
The problem is not to find a pattern, but to find the most important pages
Problems:– Important pages do not contain the search pattern
• www.porsche.com does not contain sports car or even car• www.google.com does not contain web search engine• www.airbus.com does not contain airplane
– Certain pages have nearly every word (dictionary)– Names are misleading
• http://www.whitehouse.org/ is not the web-site of the white house• www.theonion.com is not about vegetables
– Certain pattern can be found everywhere, e.g. page, web, windows, ...
Search Algorithms, WS 2004/05 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
How to rank Web-pages
The main problem about searching the web is to rank the importance
Links are very helpful:
– Humans are usually introduced on purpose
– The context of the links gives some clues about the meaning of the web-page
– Pages where many people point to are of probably very important
– Most search rely on links
Other approach: Ontology of words
– Compare the combination of words with the search word
– Good for comparing text
– Difficult if single word patterns are given
Search Algorithms, WS 2004/05 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Anatomy of a Web Search Engine
“The Anatomy of a Large-Scale Hypertextual Web Search Engine”, Sergey Brin and Lawrence Page, Computer Networks and ISDN Systems, Vol. 30, 1-6, p. 107-117, 1998
Design of the prototype– Stanford University 1998
Key components:– Web Crawler– Indexer– Pagerank– Searcher
Main difference between Google and other search engines (in 1998)
– The Pagerank mechanism
Zur Anzeige wird der QuickTime™ Dekompressor „TIFF (Unkomprimiert)“
benötigt.
Search Algorithms, WS 2004/05 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Simplified PageRank-Algorithmus
Simplified PageRank-Algorithmus
– Rank of a wep-page R(u) [0,1]
– Important pages hand their rank down to the pages they link to.
– c is a normalisation factor such that ||R(u)||1= 1, i.e.
• the sum of all page ranks add to 1
– Predecessor nodes Bu
– sucessor nodes Fu
Search Algorithms, WS 2004/05 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Simplifed Pagerank Algorithm and an example
Search Algorithms, WS 2004/05 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Matrix representaion
R c M R ,
where R is a vector (R(1),R(2),… R(n)) and M denotes the following n n – Matrix
Search Algorithms, WS 2004/05 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Simplified Pagerank Algorithm
Does it converge?
If it converges, does it converge to a single result?
Is the result reasonable?
Search Algorithms, WS 2004/05 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Eigenvector and Eigenvalue of the Matrix
For vector x and n n-matrix and a number λ:
– If M x = λ x then x is called the eigenvector and λ the eigen-value Every n n-matrix M has at most n eigenvalues
Compute the eigenvalues by eigen-decomposition
M x = λ x (M - I λ) x = 0,
where I is the identity matrix
– This equality has only non-trivial solutions if
Det(M - I λ) = 0
– This leads to a polynomial equation of degree n, which has always n solutions λ1, λ2, ..., λn
• (Fundamental theorem of algebra)
– Solving the linear equations (M - I λi) x = 0 lead to the eigenvectors
The eigenvektor of the matrix is a fix point of the recursion of the simplified pagerank algorithm
Search Algorithms, WS 2004/05 14
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Consider n discrete states and a sequence of random variable X1, X2, ... over this set of states
The sequence X1, X2, ... is a Markov chain if
A stochastic matrix M is the transition matrix for a finite Markov chain, also called a Markov matrix:
– Elements of the matrix M must be real numbers of [0, 1].
– The sum of all column in M is 1Observation for the matrix M of the simpl. pagerank algorithm
– M is stochastic if all nodes have at least one outgoing link
Stochastic Matrices
Search Algorithms, WS 2004/05 15
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The Random Surfer
Consider the following algorithm
– Start in a random web-page according to a probability distribution
– Repeat the following for t rounds
• If no link is on this page, exit and produce no output
• Uniformly and randomly choose a link of the web-page
• Follow that link and go to this web-page
– Output the web-page
Lemma
The probability that a web-page i is output by the random surfer after t rounds started with probability distribution x1, .., xn is described by the i-th entry of the output of the simplified Pagerank-algorithm iterated for t rounds without normalization.
Proof follows applying the definition of Markov chains
Search Algorithms, WS 2004/05 16
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Eigenvalues of Stochastic Matrices
Notations– Die L1-Norm of a vector x is defined as
– x0, if for all i: xi 0
– x0, if for all i: xi 0
Lemma
For every stochastic matrix M and every vector x we have
• || M x ||1 || x ||1
• || M x ||1 = || x ||1, if x0 or x0
Eigenvalues of M |i| 1
Theorem
For every stochastic matrix M there is an eigenvector x with eigenvalue 1 such that x 0 and ||x||1 = 1
Search Algorithms, WS 2004/05 17
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The problem of periodicity - Example
Search Algorithms, WS 2004/05 18
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Periodicity - Example 2
Search Algorithms, WS 2004/05 19
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Periodic Matrices
Definition– A square matrix M such that the matrix power Mk=M for k a positive integer is
called a periodic matrix.– If k is the least such integer, then the matrix is said to have period k. – If k = 1, then M2 = M and M is called idempotent.
Fact– For non-periodic matrices there are vectors x, such that limk Mk x does not
converge.
Definition– The directed graph G=(V,E) of a n x n-matrix consistis of the node set
V={1,..., n} and has edges• E = {(i,j) | Mij 0}
– A path is a sequence of edges (u1,u2),(u2,u3),(u3,u4),..,(ut,ut+1) of a graph– A graph cycle is a path where the start node is the end node– A strongly connected subgraph S is a maximum sub-graph such that every
graph cycle starting and ending in a node of S is contained in S.
Search Algorithms, WS 2004/05 20
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Necessary and Sufficient Conditions for Periodicity
Theorem (necessary condition)
– If the stochastic matrix M is periodic with period t2, then for the graph G of M there exists a strongly connected subgraph S of at least two nodes such that every directed graph cycle within S has a length of the form i t for natural number i.
Theorem (sufficient condition)
– Let the graph consist of one strongly connected subgraph and
– let L1,L2, ..., Lm be the lengths all directed graph cycles of maximal length n
– Then M is non-periodic if and only if gcd(L1,L2, ..., Lm) = 1
Notation:
– gcd(L1,L2, ..., Lm) = greatest common divisor of numbers L1,L2, ..., Lm
Corollary
– If the graph is strongly connected and there exists a graph cycly of length 1 (i.e. a loop), then M is non-periodic.
Search Algorithms, WS 2004/05 21
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Disadvantages of the Simplified Pagerank-Algorithm
The Web-graph has sinks, i.e. pages without links
M is not a stochastic matrix
The Web-graph is periodic Convergence is uncertain
The Web-graph is not strongly connected Several convergence vectors possible
Rank-sinks – Strongly connected subgraphs absorb all weight of the predecessors – All predecessors pointing to a web-page loose their weight.
Search Algorithms, WS 2004/05 22
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
The (non-simplified) Pagerank-Algorithm
Add to a sink links to all web-pages
Uniformly and randomly choose a web-page
– With some probability q < 1 perform a step of the simplified Pagerank algorithm
– With probability 1-q start with the first step (and choose a random web-page)
Note M ist stochastic
Search Algorithms, WS 2004/05 23
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Properties of the Pagerank-Algorithm
Graph der Matrix is strongly connected
There are graph cycles of length 1
Theorem
In non-periodic matrices of strongly connected graphs the Markov-chain converges to a unique eigenvector with eigenvalue 1.
PageRank converges to this unique eigenvector
24
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Thanks for your attentionEnd of 6th lectureNext lecture: Mo 29 Nov 2004, 11.15 am, FU 116
Next exercise class: Mo 22 Nov 2004, 1.15 pm, F0.530 or We 24 Nov 2004, 1.00 pm, E2.316