An applica)on of Markov Chains: PageRank. Finding relevant informa)on on the Web
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1 An applica)on of Markov Chains: PageRank Finding relevant informa)on on the Web
2 Please Par)cipate h>p://
3
4 How much do you know about PageRank? 1) Nothing. 2) I heard about it, but don t know what it is 3) I learned about it but I forgot 4) I know it well
5 Key Ques)on: Best Answer (thus far): Best Part: find relevant informa)on in this large- scale distributed system PageRank you know the tools to appreciate it!
6 Map of Internet 2003 h>p:// web: 1 trillion pages - 1 trillion links brain: 0.1 trillion neurons trillion links
7 TU DelC: 8 ES group: 5
8 Ques)on 1) Which page is the most relevant? This is a good university This is a great university This is a university Answers: 1) page A 2) page B 3) page C 4) All
9 Ques)on 2) Which page is the most relevant, now? This is a good university This is a great university This is a university Answers: 1) page A 2) page B 3) page C 4) All
10 The Story global search share (%) Problem: how to find relevant informa;on? PageRank Algorithm Google Yahoo! ) PageRank Two CS PhD students 1) Markov insight 2) Before PageRank 4) ACer PageRank 5) Bracelet experiment
11 1) The Markov Insight: on probability, free- will and poetry 1713: Independent events [Bernoulli] Gambling: coin flipping and dice rolling steady state: weak law of large numbers 1902: Free- will [Nekrasov] theory used to prove free- will (over predes)na)on). 1906: Dependent events [Markov] Markov didn t like this abuse of mathema)cs steady state under some condi)ons: used a poem as test case p(vowel) = 0.43 Independent: p(vowel vowel) = 0.43^2 = 0.19 Dependent: p(vowel vowel) = 0.06 Nice ar)cle: h>p:// links- in- the- markov- chain/
12 Example Localiza)on Your posi)on at t+1 depends on where you were at t. You don t teletransport! There are many many applica)ons, stock market, social sciences, biology, gene)cs, voice recogni)on, etc
13
14 So what What does the Markov insight has to do with PageRank? Everything Your Turn!
15 2) Life Before PageRank
16 Imperfect Solu)on 1: Word Frequency - problem This is a great University. Our University provides the best educa)on. No other University matches our strength. Our University has a beau)ful campus.
17 Imperfect Solu)on 1: Word Frequency - outcome President of Stanford did his own search
18 But the WWW is a graph Wouldn t it make sense to look at the graph structure? But what characteris)cs of the graph are the ones what ma>er?
19 Ques)on 3) A graph consists of nodes connected via links: incoming and outgoing links. We will see that both play a major role in PageRank, but for now use your intui)on to iden)fy the most relevant source: A B C D F Answers: 1) page A 2) page B 3) page D 4) page F E
20 Imperfect Solu)on 2: Incoming Links - metric Which orange node is more relevant? example: research papers centrality = degree(v)/ E problem of cita)ons
21 Imperfect Solu)on 2: Incoming Links - problem reputa)on of the source ma>ers centrality = degree(v)/ E reputa)on = Σ r(j), where j are neighbors
22 Outgoing Links Incoming links ma>er Reputa)on of sources ma>ers What about outgoing links?
23 Outgoing Links: set up
24 Outgoing Links: Problem not only ma>ers reputa)on of source, but also if we are special centrality = degree(v)/ E reputa)on = Σ r(j) r(i) = Σ ( r(j) / N(j) )
25 Ques)on 4) Recall: incoming links ma>er, outgoing links ma>er, reputa)on ma>ers. Assume that ini)ally all nodes have the same reputa)on. Which node is the most relevant? C Answers: A B D F 1) page A 2) page B 3) page D 4) page F E
26 Markov Chains to the rescue! Incoming links ma>er Reputa)on ma>ers Outgoing links ma>er r(i) = Σ ( r(j) / N(j) ) S)ll, how do we get reputa)ons?
27 Reputa)ons look like a Markov Chain.Remember the water filling slide?
28 Ques)on 5) What is the node with the highest page rank? And what is its rank? Answers: 1) page A 2) page B 3) page C 4) pages B and C Answer format: (page X, rank Y);(page Z, rank R); Got stuck? State where! Calculator: h>p://wims.unice.fr/wims/en_tool~linear~linsolver.en.html
29 Solu)on 5) Linear System a = c/2 b = a + c/2 c = b a + b + c = 1 Solu)on: a=0.2, b=0.4, c=0.4
30 What are the problems with this Naïve Page Rank? You are given the web graph and are asked to run the Markov Chain you know. What problems do you foresee occurring? Your turn!
31 Irreducibility Absorp)on states Graph par))ons Steady state (Rank) depends on inputs How do we solve it? C A B D F G H E
32 The Story global search share (%) Theore)cally there were other seminal studies. Authorita)ve Sources in a Hyperlinked Environment, Jon M. Kleinberg, Google Yahoo! ) PageRank 1) Markov insight 2) Before PageRank 4) ACer PageRank
33 Damping factor Create virtual connec)ons to all other nodes. Weight of new connec)ons (red) is lower than original ones (blue) C A B D F G H E
34 Ques)on 6) What is the effect of adding links among all nodes to the final steady state (rank)? 1) The reputa)ons will be more spread out (capitalism). 2) The reputa)ons will be more similar (socialism). 3) The reputa)ons will remain the same (no effect) C A B D F G H E
35 Recap Incoming links ma>er Reputa)on ma>ers Outgoing links ma>er These graph proper)es can be combined with Markov Chains to obtain the ranks of pages (steady state) But applying blindly MC have problems. Is the WWW graph irreducible and aperiodic? The damping factor overcomes the irreducibility problem. Let s do an example
36 Damping factor C A B D F G H E
37 Itera)ve Solu)on The PageRank cita)on ranking: Bringing order to the web. L Page, S Brin, R Motwani, T Winograd, 1999
38 The PageRank or Google Matrix
39 Ques)on 7) Assume a damping factor of p = 0.3. Find the nodes ranks? Answer format: (page X, rank Y);(page Z, rank R); Matrix calculator: h>p://matrix.reshish.com/ h>p://wims.unice.fr/wims/en_tool~linear~linsolver.en.html
40 Solu)on matrix and rank p = 0.3, n=3, hence B = 0.3/ A = M Rank = ~{a=0.23, b=0.39, c=0.37} w/o damping factor: a=0.2, b=0.4, c=0.4 why is this happening?
41 First a simple test samplematrixmul)plica)on.m Setup = 1, df = 0 & df = 0.15; Setup = 2, df = 0 & df =0.15 Setup = 3, df = 0 & df = 0.15; What s the problem (discussion board) setup = 1 setup = 2 setup = 3
42
43 Ques)on 8) Matrix calculator: h>p://matrix.reshish.com/
44 p=0.4
45 Periodicity Problem is that there is no steady state How do you solve it? is guaranteed in prac)ce for the Web Topic- Sensi)ve PageRank, Taher H. Haveliwala, WWW Page 2 column 2
46
47 Experimental data matrices from experiments 1) Check if matrix is column stochas)c and irreducible 2) (If needed) Make matrix irreducible. Add self links 3) (if needed) Normalize matrix: column stochas)c. 4) Calculate B matrix (all ones). Damping factor = ) Obtain M matrix, M = p*b + (1- p)a 6) Obtain the steady state 7) Enumerate the top- 5 social bu>erflies.
48 Data descrip)on Proximity: RSS Time of contact: Number of periods Directed graphs (friendship is not mutual) Data depic)on graph_- 50_0_lunch graph_- 50_50_lunch
49 Data is anonymous lunch: 60 par)cipants drinks: 30 par)cipants Active IDs on each experiment drinks lunch Connec)vity Matrix
50 Recap Markov insight, 1906 WWW, 1990 s First search engines based on word frequency Proper)es of graph are important PageRank, 1998 A Markov Chain with a damping factor Personalized PageRank
51 Ques)on 10) Compute the PageRank assume a damping constant p =0.2
52 Solu)on Matrix & Rank p=0.2, n=8, B = [ 0.2/8 = 0.025], M Rank = { 0.725, 0.065, 0.085, 0.025, 0.025, 0.025, 0.025, 0.025}
53 Links The PageRank cita)on ranking: Bringing order to the web. L Page, S Brin, R Motwani, T Winograd, 1999, Stanford University h>p://ilpubs.stanford.edu:8090/422/1/ pdf Networked Markets. Ashish Goel, Stanford University h>p://web.stanford.edu/class/msande233/handouts/lectures6-7.pdf The Mathema)cs of Web Search. Raluca Tanase and Remus Radu, Cornell University. h>p:// lecture3.html PageRank Examples. Dell Zhang, University of London h>p:// Topic- Sensi)ve PageRank. Taher H. Haveliwala, Stanford University h>p://research.taherh.org/pubs/topic- sensi)ve- pagerank.pdf
54 or Subject QEES pra)cum
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