1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Search Algorithms Winter Semester 2004/ Nov th Lecture Christian Schindelhauer

Search Algorithms, WS 2004/05 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Chapter III Searching the Web 22 Nov 2004

Search Algorithms, WS 2004/05 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Overview Search Engines (March 2002)  Number of documents Search Engine Showdown Estimate (millions) Claim (millions) Google9681,500 WiseNut5791,500 AllTheWeb Northern Light AltaVista Hotbot MSN Search

Search Algorithms, WS 2004/05 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Overview Search Engines (Dez. 2002)  Number of documents Search Engine Showdown Estimate (millions) Claim (millions) Google3,0333,083 AlltheWeb2,1062,116 AltaVista1,6891,000 WiseNut1,4531,500 Hotbot1,1473,000 MSN Search1,0183,000 Teoma1, NLResearch Gigablast275150

Search Algorithms, WS 2004/05 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Problems of Searching the Web  Currently (Nov 2004) more than 8 billion = millions web-pages – 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 does not contain sports car or even carwww.porsche.com does not contain web search enginewww.google.com does not contain airplanewww.airbus.com –Certain pages have nearly every word (dictionary) –Names are misleading is not the web-site of the white househttp:// is not about vegetableswww.theonion.com –Certain pattern can be found everywhere, e.g. page, web, windows,...

Search Algorithms, WS 2004/05 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 , 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

Search Algorithms, WS 2004/05 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 B u –sucessor nodes F u

Search Algorithms, WS 2004/05 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Simplifed Pagerank Algorithm and an example

Search Algorithms, WS 2004/05 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 λ 1, λ 2,..., λ n (Fundamental theorem of algebra) –Solving the linear equations (M - I λ i ) 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer  Consider n discrete states and a sequence of random variable X 1, X 2,... over this set of states  The sequence X 1, X 2,... 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 1  Observation 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 x 1,.., x n 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Eigenvalues of Stochastic Matrices  Notations –Die L1-Norm of a vector x is defined as –x  0, if for all i: x i  0 –x  0, if for all i: x i  0  Lemma For every stochastic matrix M and every vector x we have || M x || 1  || x || 1 || M x || 1 = || x || 1, if x  0 or x  0  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 INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The problem of periodicity - Example

Search Algorithms, WS 2004/05 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Periodicity - Example 2

Search Algorithms, WS 2004/05 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Periodic Matrices  Definition –A square matrix M such that the matrix power M k =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 M 2 = M and M is called idempotent.  Fact –For non-periodic matrices there are vectors x, such that lim k  M k 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) | M ij  0} –A path is a sequence of edges (u 1,u 2 ),(u 2,u 3 ),(u 3,u 4 ),..,(u t,u t+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 INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Necessary and Sufficient Conditions for Periodicity  Theorem (necessary condition) –If the stochastic matrix M is periodic with period t  2, 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 L 1,L 2,..., L m be the lengths all directed graph cycles of maximal length n –Then M is non-periodic if and only if gcd(L 1,L 2,..., L m ) = 1  Notation: –gcd(L 1,L 2,..., L m ) = greatest common divisor of numbers L 1,L 2,..., L m  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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian 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 INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention End of 6th lecture Next lecture:Mo 29 Nov 2004, 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