Calcolo di PageRank in Google mediante un algoritmo Las Vegas e riflessioni sui metodi randomizzati per sistemi complessi

Transcription

1 Calcolo di PageRank in Google mediante un algoritmo Las Vegas e riflessioni sui metodi randomizzati per sistemi complessi Roberto Tempo CNR-IEIIT Consiglio Nazionale delle Ricerche Politecnico di Torino tempo@polito.it

2 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

3 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

4 Randomized Algorithm Randomized Algorithm (RA): An algorithm that makes random choices during its execution to produce a result Example: Matlab code set_r=1:0.01:3; 01:3; for k=1:length(set_r) if (FRA(r) > 0) a_opt(k) = rand; a_lin(k)=(e/(e-1))*r; a_sqr(k)=r+sqrt(2*(r-1)); a=a_sqr(k); a_sub(k)=(a/(a-1))*(r+log(a)-1); end

5 Randomized Algorithms (RAs) Randomized algorithms are frequently used in many areas of engineering, computer science, physics, finance, optimization, but their hi appearance in systems and control is mostly limited to Monte Carlo simulations

6 Randomized Algorithms (RAs) Computer science (RQS for sorting) Mathematics of finance (expected value computation) Distributed algorithms (PageRank in Google) Robotics (motion and path planning problems) Bioinformatics (string matching problems) Computer vision (computational geometry)

7 A Success Story: Randomization in Computer Science

8 A Success Story in CS Problem: Sorting N real numbers Algorithm: RandQuickSort (RQS) RQS is implemented in a C library of Linux for sorting numbers C.A.R. Hoare (1962) D.E. Knuth (1998)

9 A Success Story in CS Problem: Sorting N real numbers Algorithm: RandQuickSort (RQS) RQS is implemented in a C library of Linux for sorting numbers Sorting Problem given N real x 1 x 2 x 3 sort them in numbers x 4 x 5 x 6 increasing order S 1

10 RandQuickSort (RQS) TheideaistodividetheoriginalsetS 1 into two sets having (approximately) the same cardinality This requires finding the median of S 1 (which may be difficult) This operation is performed using randomization

11 RandQuickSort (RQS) RQS is a recursive algorithm consisting of two phases 1. randomly select a number x i (e.g. x 4 ) 2. deterministic comparisons between x i and other (N-1) numbers x 2 x 3 x 1 x 5 x 6 x 4 S 2 S 3 numbers smaller than x 4 numbers larger than x 4

12 RQS: Binary Tree Structure We use randomization at each step of the (binary) tree

13 Running Time of RQS Because of randomization, running time may be different from one run of the algorithm to the next one RQS is very fast: Average running time is O(N log(n)) Major improvement compared to brute force approach O(N 2 )whenn =2 M Average running time holds for every input with probability at least 1-1/N (i.e. it is highly probable)

14 (Quasi) Monte Carlo Methods for Computational Finance QMC methods to estimate the prize of collaterized mortgage obligations The problem is to approximate the average mortgage [0,1] f ( u )d u n with its empirical mean (based on random samples u i ) N 1 f ( ui ) N Curse of dimensionality: n = 360! i 1

15 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

16 The PageRank Problem A Systems and Control Viewpoint The PageRank Problem is useful for developing novel ideas within systems and control

17 The PageRank Problem A Systems and Control Viewpoint Technical Tool: Theory of Stochastic Matrices

18 PageRank PageRank is used in search engines to indicate the importance of the page currently visited PageRank has broader utility than search engines Used in various areas for ranking other objects o scientific journals (Eigenfactor) o e-commerce and e-consultancy o cancer biology and protein detection o ranking top tennis players (Jimmy Connors #1) o

19 Website of Umberto Eco Looking for Umberto Eco in Google we find immediately the website

20 PageRank for Umberto Eco Using a PageRank checker we compute PageRank is Google s view of the importance of this page PageRank is a numerical value in the interval (0,1] which indicates the importance of the page you are visiting

21 Graphs 1 2 Directed graph with nodes and links 3 4 5

22 Random Web Surfer Model Consider a set of pages (nodes) connected by directed communication links Web surfer moves along randomly following the hyperlink structure When arriving at a page with several outgoing links, one is chosen at random, then the random surfer moves to a new page, and so on

23 Random Web Surfer Model Web representation with incoming and outgoing links

24 Random Web Surfer Model

25 Random Web Surfer Model Pick an outgoing link at random

26 Random Web Surfer Model Arriving at a new web page

27 Random Web Surfer Model Pick another outgoing link at random

28 Random Web Surfer Model

29 Random Web Surfer Model

30 Random Web Surfer Model

31 Random Web Surfer Model Ifapageisimportant then it is visited more often... The time the random surfer spends on a page is a measure of its importance If important pages point to your page, then your page becomes important because it is visited often What is the probability that your page is visited? Need to rank the pages in order of importance for facilitating the web search

32 Graph Representation Directed graph with nodes (pages) and links representing the web Graph is not necessarily strongly connected (from 5 you cannot reach other pages) Graph is constructed using crawlers, spiders, sniffers moving continuously along the web

33 Hyperlink Matrix For each node we count the number of outgoing links and normalize their sum to 1 Hyperlink matrix is a nonnegative (column) substochastic matrix 5

34 Hyperlink Matrix / /2 0 A 1/ /2 0 1/

35 PageRank: Bringing Order to the Web We rank n web pages in order of importance PageRank x* of the hyperlink matrix A is defined as x*=ax* where x* [0,1] n and i x i *=1 x* is a nonnegative unit eigenvector corresponding to the eigenvalue 1 of A S. Brin, L. Page (1998) S. Brin, L. Page, R. Motwani, T. Winograd (1999)

36 PageRank: Bringing Order to the Web We rank n web pages in order of importance PageRank x* of the hyperlink matrix A is defined as x*=ax* where x* [0,1] n and i x i *=1 x* is a nonnegative unit eigenvector corresponding to the eigenvalue 1 of A The question is when x* exists and it is unique

37 Issue of Black Holes First issue: We have black holes (pages having no outgoing link) Random surfer gets stuck when visiting a pdf file In this case the back button of the browser is used Mathematically, the hyperlink matrix is nonnegative and (column) substochastic Easy fix: Add new links to make the matrix stochastic

38 Page 5 is a Black Hole / /2 0 A 1/ /2 0 1/

39 1 2 Easy Fix: Add New Link We add a new outgoing link from page 5 to page A / /2 0 1/ /2 1 1/

40 In General the Fix is not so Easy 1 2 Page 5 has two incoming links 3 4 5

41 In General the Fix is not so Easy 1 2 We add an outgoing link from 5 to A / /3 0 1/ /3 1 1/ /3 0

42 In General the Fix is not so Easy 1 2 or we add an outgoing link from 5 to 4? A / /3 0 1/ /3 0 1/ /3 0

43 Modified Hyperlink Matrix A solution may be to break page 5 into two pages 5a and 5b This artificially changes the number of pages (not only the number of links) The topology of the network changes 5a 5b

44 Modified Hyperlink Matrix / / / /3 1 0 A 4 1/ / a 5b

45 Assumption: No Black Holes This is a web modeling problem Assume that there are no black holes This implies that A is a nonnegative stochastic matrix (instead of substochastic) having at least one eigenvalue equal to one Second issue: This eigenvalue is not necessarily unique

46 Teleportation Matrix Teleportation: After a while the random surfer gets bored and decides to jump over long distance to another page not directly connected to that currently visited New pagemay pg be geographically g located far away

47 Recall the Random Web Surfer Model Web representation with incoming and outgoing links

48 Recall the Random Web Surfer Model

49 Recall the Random Web Surfer Model

50 Recall the Random Web Surfer Model

51 Recall the Random Web Surfer Model

52 Teleportation Model We are teleported to a web page located far away

53 Random Web Surfer Model Again Pick another outgoing link at random

54 Random Web Surfer Model Again Pick another outgoing link at random

55 Teleportation Model Again We are teleported to another web page located far away

56 Convex Combination of Matrices Teleportation model is represented as a convex combination of matrices A and S/n 1 1 S = 11 T is a rank-one matrix S 1 vector with all entries equal to one 1 1 Consider a matrix M defined as M =(1-m) A + m/n S m (0,1) where n is the number of pages The value m = 0.15 is used at Google

57 Matrix M M is a convex combination of two nonnegative stochastic matrices and m (0,1) M is a positive stochastic matrix

58 Properties of Matrix M and Perron Theorem Matrix M is primitive (M k is positive for some k) M is irreducible and the corresponding graph is strongly connected (every page is connected to every page) The eigenvalue 1 is a simple eigenvalue of maximum modulus The corresponding eigenvector is positive

59 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

60 Monte Carlo and Las Vegas Monte Carlo was invented by Metropolis, Ulam, von Neumann, Fermi, (Manhattan project) Metropolis Fermi Ulam, Feymann, von Neumann Las Vegas first appeared in computer science in the late seventies

61 Randomized Algorithm Randomized Algorithm (RA): An algorithm that makes random choices during its execution to produce a result Example: Matlab code set_r=1:0.01:3; 01:3; for k=1:length(set_r) if (FRA(r) > 0) a_opt(k) = rand; a_lin(k)=(e/(e-1))*r; a_sqr(k)=r+sqrt(2*(r-1)); a=a_sqr(k); a_sub(k)=(a/(a-1))*(r+log(a)-1); end

62 Monte Carlo Randomized Algorithm Monte Carlo Randomized Algorithm (MCRA): A randomized algorithm that may produce incorrect results, but with bounded probability of error Prob{error > } <2e (-2N2 ) Hoeffding inequality where is the probabilistic accuracy of the estimate, N is the sample size (sample complexity) and e is the Euler number

63 Example of Monte Carlo: Area/Volume Estimation Estimate the volume of the red area: Generate N samples uniformly in the rectangle (hint: use Matlab rand) Count how many fall within the red area (M), estimated area = M/N

64 Homework Using rand we can generate N samples in a rectangle Question: how can we generate N samples in a sphere? 1 step RACT: Randomized Algorithms Control Toolbox for Matlab

65 Las Vegas Randomized Algorithm Las Vegas Randomized Algorithm (LVRA): A randomized algorithm that always produces correct results, the only variation from one run to another is the running time Example: Randomized Quick Sort (RQS)

66 Example of Las Vegas: Discrete Random Variables Consider discrete random variables q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10

67 Example of Las Vegas: Discrete Random Variables Consider discrete random variables q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10

68 Example of Las Vegas: Discrete Random Variables Consider discrete random variables q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10

69 Parallel and Distributed Simulations Random samples are independent identically distributed This approach leads to parallel and distributed simulations IBM Blue Gene Cray-1 vector processor

70 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

71 PageRank Computation with Power Method PageRank is computed with the power method x(k+1) = Mx(k) If i x i (0) = 1 convergence is guaranteed by Perron Theorem because M is a primitive matrix x(k) x* for k Remark: PageRank computation may be interpreted as finding the stationary distribution of a Markov Chain (steady-state probability that page i is visited is x* ) i

72 PageRank Computation with Power Method / /2 1/3 A 0 1/2 0 1/3 0 1/2 1/2 0 m M x* T

73 Size of the Web ThesizeofM is more than 8 billion (and it is increasing)! The PageRank computation requires iterations This computation takes about a week and it is performed centrally at Google once a month More and more computing power is needed

74 Columbia River, The Dalles, Oregon Land is cheap Electricity is cheap Water of the river may be used as a cooling device

75 Randomized d Decentralized Algorithms

76 Randomized Decentralized Algorithms for PageRank Computation Main idea: Develop Las Vegas randomized decentralized algorithms for computing PageRank H. Ishii, R. Tempo, Distributed Randomized Algorithms for the PageRank Computation, IEEE TAC, 2010

77 Decentralized Communication Protocol 1 4 Gossip communication protocol: at time k randomly select page i (i=4) for PageRank update 2 3

78 Decentralized Communication Protocol 1 4 Gossip communication protocol: at time k randomly select page i (i=4) for PageRank update 2 3

79 Decentralized Communication Protocol 1 4 Gossip communication protocol: at time k randomly select page i (i=4) for PageRank update send PageRank value of page i to the outgoing pages that are linked to page i

80 Decentralized Communication Protocol 1 4 Gossip communication protocol: at time k randomly select page i (i=4) for PageRank update send PageRank value of page i to the outgoing pages that are linked to page i 2. request PageRank values from incoming pages that are linked to page i

81 Las Vegas Randomized Approach The pages taking action are determined via a stochastic process θ(k) {1,, n} θ(k) is assumed to be i.i.d. with uniform probability Prob{θ(k)=i} = 1/n If at time k θ(k) =i then page i initiates PageRank update

82 Randomized Update Scheme Consider the randomized update scheme x(k+1) = A θ(k) x(k) where A θ(k) are the decentralized link matrices (example next) Study average properties of A θ(k) Define the time average k 1 yk ( ) xi ( ) k 1 i0

83 Decentralized Link Matrices - 1 A / /2 1/3 0 1/2 0 1/3 0 1/2 1/2 0 A 4 1/3 1/3 1/3 0 1/2 1/2 0

84 Decentralized Link Matrices - 2 A / /2 1/3 0 1/2 0 1/3 0 1/2 1/2 0 A / / /3 0 1/2 1/2 0

85 Decentralized Link Matrices - 3 A / /2 1/3 0 1/2 0 1/3 0 1/2 1/2 0 A /3 0 1/2 0 1/ /2 1/3 0 1/2 1/2 0

86 Decentralized Link Matrices - 4 A / /2 1/3 0 1/2 0 1/3 0 1/2 1/2 0 A /2 1/ /2 0 1/ /2 2/3 A /3 0 1/2 0 1/ /2 1/3 0 1/2 1/2 0

87 Modified Randomized Update Scheme Recall: Need to work with positive stochastic matrices (teleportation model) Consider the modified randomized update scheme x(k+1) = M θ(k) x(k) where M θ(k) are the modified decentralized link matrices computed as M i =(1-r) A i + r/n S i=1,2,,n and r (0,1) is a design parameter r =2m/(n mn +2m)

88 Convergence Properties Theorem (convergence properties) The time average y(k) of the modified randomized update scheme converges to PageRank x* inmse E[ y(k) -x* 2 ] 0 for k provided that i x i (0) = 1 Proof: Based on the theory of ergodic matrices H. Ishii, R. Tempo (2010)

89 Comments Time average y(k) can be computed recursively as a function of y(k-1) Because of averaging convergence rate is 1/k Sparsity of the matrix A is preserved because x(k+1) = Mx(k) =(1-r) Ax(k) +r/n 1 where 1 is a vector with all entries equal to one A / / 2 1 / / / / 2 1 / M=

90 Extensions Stopping criteria to compute approximately PageRank Extensions to simultaneous update of multiple web pages Robustness for fragile links Convergence results for update schemes based on outgoing links (not incoming)

91 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

92 ISI Web of Knowledge

93 Ranking Journals: Impact Factor Impact Factor IF IF 2010 number citations in 2010 of articles published in number of articles published in Remark: Impact Factor is a flat criterion (it does not take into account where the citations come from)

94 ISI Web of Knowledge

95 Ranking Journals: Eigenfactor Eigenfactor EF Ranking journals using ideas from PageRank In Eigenfactor journals are considered influential if they are cited often by other influential journals What is the probability that a journal is cited? C. T. Bergstrom (2007)

96 Article Vector Article vector v v i number articles published by journal i in number articles published by all journals in v i is the fraction of all published articles coming from journal i during the period Article vector v is a stochastic vector Total number of journals = 7611

97 Cross-citation Matrix - 1 Aij number citations in 2010 from journal j to articles published in journal i in Self-citations are omitted 0 A 0 0 Normalization to obtain a column substochastic matrix Aij Aij Akj (note: if 0/0 we set A ij =0) k

98 Cross-citation Matrix - 2 Black holes: journals that do not cite any other journal Columns with entries equal to zero A substochastic matrix A Replace A with stochastic matrix introducing the vector v A v

99 Journal Influence Vector Consider the Eigenfactor equation (same form of the PageRank equation) M 1 m A m v m T ( ) x* isthejournal influence vector (defined as PageRank for M) which provides the weights Eigenfactor EF is defined as 100 Ax* EF ( Ax*) i i

100 Comments - 1 In Eigenfactor equation we replace the rank-one matrix 1/nS= 1/n 11 T with the rank-one weighted matrix v 1 T 1/n S 1/ n 1/ n 1/ n 1/ n M is a positive stochastic matrix v v v 1 1 T 1 v Journal influence vector x* exists and it is unique It is the eigenvector corresponding to the simple eigenvalue 1 (from Perron Theorem) n v n

101 Comments - 2 Journal influence vector x* is the steady-state fraction of time spent visiting each journal in M Eigenfactor EF j of journal j is the percentage of the total weighted 100 Ax* EF citations that journal j receives ( Ax*) i from all 7611 journals Weights given by x* (with no weights we obtain IF) i

102 PageRank and Twitter Followers

103 PageRank and Twitter Followers

104 References

105 PageRank Journal References H. Ishii, R. Tempo, E.-W. Bai, PageRank and Web aggregation, IEEE TAC, summer 2012 O. Fercoq, M. Akian, M. Bouhtou, S. Gaubert, PageRank, Markov Chains and Ergodic Control, IEEE TAC, to appear 2012 W.-X. Zhao, H.-F. Chen, H.T. Fang, Almost-Sure Convergence of the Randomized Scheme, 2012 C. de Kerchove, L. Ninove, and P. Van Dooren. Influence of the Outlinks in PageRank. LAA, 2008 A.V. Nazin and B.T. Polyak. Randomized Algorithm to Determine the Eigenvector of a Stochastic Matrix, Automation and Remote Control, 2011

106 References on Randomized Algorithms G. Calafiore, F. Dabbene and R. Tempo Research on Probabilistic Design Methods, Automatica, 2011 [1304 downloads during March-December 2011, #2 in the ranking] F. Dabbene and R. Tempo, Probabilistic and Randomized Tools for Control Design, g, The Control Handbook (W. S. Levine Ed.), Taylor & Francis, 2010 R. Tempo, G. Calafiore and F. Dabbene, Randomized Algorithms for Analysis and Control of Uncertain Systems, Springer-Verlag, London, 2005 (second edition to appear in 2012)

107 List of Topics o o o o o o Randomization in CS and Math Finance The PageRank Problem in Google Monte Carlo and Las Vegas (Distributed) Algorithms PageRank Computation Ranking Scientific Journals Conclusions

108 Where is the Beef? Where is the controller? Question that the late Peter Dorato posed several times when I was studying control problems without a controller

109 Consensus and PageRank Consider a set of agents each having a numerical value transmitted to the neighboring agents iteratively Consensus: All agents eventually reach a common value Consensus of multi-agent systems and PageRank computation share striking similarity properties