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PageRank all the things! @arnoutboks Arnout Boks #dpc19 08-06-2019

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@arnoutboks #dpc19 Story time

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@arnoutboks #dpc19 Story time # Team Pnt 1 DEO 2 30 2 Netwerk 1 24 3 Punch 6 24 4 Delta 5 23 5 Red Stars 1 23 6 Delta 4 17 7 Kalinko 6 12 8 Kratos 7 7 9 Punch 9 6 10 Sovicos 4 4 next match: Netwerk 1 – Delta 4

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@arnoutboks #dpc19 Story time # Team Pnt 1 DEO 2 30 2 Netwerk 1 24 3 Punch 6 24 4 Delta 5 23 5 Red Stars 1 23 6 Delta 4 17 7 Kalinko 6 12 8 Kratos 7 7 9 Punch 9 6 10 Sovicos 4 4 Can we account for the fact that some teams have yet played against weaker opposition than others?

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@arnoutboks #dpc19 To be continued… $ git stash Saved working directory and index state WIP on master: 5002d47 PageRank all the things! HEAD is now at 5002d47 PageRank all the things!

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@arnoutboks #dpc19 PageRank

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Linear Algebra Some necessary math

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@arnoutboks #dpc19 Linear Algebra is the discipline that studies vectors and matrices

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@arnoutboks #dpc19 Vectors v = 8 3 -4 1

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@arnoutboks #dpc19 Vectors v = 8 3 -4 1 4 dimension 4 (“4-vector”)

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@arnoutboks #dpc19 Vectors v = -π 4.2 2 dimension 2 (“2-vector”)

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@arnoutboks #dpc19 Vectors as coordinates 1 1 1 -2 -3 -2 x y

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@arnoutboks #dpc19 Scalar multiplication v = -1.5 1 3v = -1.5 1 3 = -4.5 3

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@arnoutboks #dpc19 Scalar multiplication x y v 3v

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@arnoutboks #dpc19 Matrices 8 0 42 3 3 7 4 -7 1 1 2 6 M =

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@arnoutboks #dpc19 Matrices 8 0 42 3 3 7 4 -7 1 1 2 6 M = 4 dimension 4, 3 (“4x3-matrix”) 3

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@arnoutboks #dpc19 Matrix-vector multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 M v 8 3 -4

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@arnoutboks #dpc19 Matrix-vector multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 M 3 v 8 3 -4 3

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@arnoutboks #dpc19 Matrix-vector multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 M v 8 3 -4 Mv =

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@arnoutboks #dpc19 Matrix-vector multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 M v 8 3 -4 8 3 -4 Mv =

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@arnoutboks #dpc19 Matrix-vector multiplication Mv = 8 x 8 0 x 3 42 x -4 3 x 8 3 x 3 7 x -4 4 x 8 -7 x 3 1 x -4 1 x 8 2 x 3 6 x -4 8 3 -4

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@arnoutboks #dpc19 Matrix-vector multiplication Mv = 64 0 -168 24 9 -28 32 -21 -4 8 6 -24

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@arnoutboks #dpc19 Matrix-vector multiplication Mv = 64 + 0 + -168 24 + 9 + -28 32 + -21 + -4 8 + 6 + -24

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@arnoutboks #dpc19 Matrix-vector multiplication Mv = 64 + 0 + -168 24 + 9 + -28 32 + -21 + -4 8 + 6 + -24 = -104 5 7 -10

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@arnoutboks #dpc19 Matrix-vector multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 8 3 -4 Mv = = -104 5 7 -10

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@arnoutboks #dpc19 Matrix-vector multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 8 3 -4 Mv = = -104 5 7 -10 “4x3-matrix multiplied by a 3-vector yields a 4-vector”

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The Matrix In more detail

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@arnoutboks #dpc19 Matrix as a transformation 8 3 -4 -104 5 7 -10 M 4x3-matrix 3D 4D

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@arnoutboks #dpc19 Chaining transformations M 4x3-matrix 4D 3D N 2x4-matrix 2D

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@arnoutboks #dpc19 Chaining transformations M 4x3-matrix 4D 3D N 2x4-matrix 2D

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@arnoutboks #dpc19 Chaining transformations M 4x3-matrix 4D 3D N 2x4-matrix 2D Is there a 2x3-matrix describing this transformation?

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@arnoutboks #dpc19 Matrix multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 NM = 1 0 -2 1 3 1 0 -1

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@arnoutboks #dpc19 Matrix multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 NM = 1 0 -2 1 3 1 0 -1 8 3 4 1 = 1 27

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@arnoutboks #dpc19 Matrix multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 NM = 1 0 -2 1 3 1 0 -1 0 3 -7 2 = 1 16 27 1

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@arnoutboks #dpc19 Matrix multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 NM = 1 0 -2 1 3 1 0 -1 42 7 1 6 = 1 16 46 27 1 127

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@arnoutboks #dpc19 Matrix multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 NM = 1 0 -2 1 3 1 0 -1 = 1 16 46 27 1 127

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@arnoutboks #dpc19 Matrix multiplication 8 0 42 3 3 7 4 -7 1 1 2 6 NM = 1 0 -2 1 3 1 0 -1 = 1 16 46 27 1 127 “2x4-matrix multiplied by 4x3-matrix yields a 2x3-matrix”

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@arnoutboks #dpc19 Matrix multiplication as chained transformation M 4x3-matrix 4D 3D N 2x4-matrix 2D 3D NM 2x3-matrix 2D N(Mv) = (NM)v

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@arnoutboks #dpc19 Square matrices 2D M 2x2-matrix 2D

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@arnoutboks #dpc19 Square matrices 2D M 2x2-matrix 2D

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@arnoutboks #dpc19 Square matrices x y v Mv 0 -1 1 0 M =

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@arnoutboks #dpc19 Square matrices x y v Mv 1 0 0 -1 M =

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@arnoutboks #dpc19 Square matrices x y v Mv 1 0 0 -1 M = u = Mu

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PageRank Ranking web search results

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@arnoutboks #dpc19 Web pages and links A B

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@arnoutboks #dpc19 Web pages and links A B

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@arnoutboks #dpc19 Web pages and links A B C D

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@arnoutboks #dpc19 Web pages and links B E C F D A G

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@arnoutboks #dpc19 The PageRank of a page depends on the PageRank of the pages linking to it

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@arnoutboks #dpc19 Chicken and egg problem

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@arnoutboks #dpc19 Approach Let n be the number of web pages Let s be an n-vector of scores for these pages Let M be an n×n-matrix describing the dependencies of scores

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@arnoutboks #dpc19 Approach Let n be the number of web pages Let s be an n-vector of scores for these pages Let M be an n×n-matrix describing the dependencies of scores s = Ms

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@arnoutboks #dpc19 Creating the matrix M A B C D A 0 0 0 0 B 0 0 0 0 C 0 0 0 0 D 1 0 0 0 Outbound 1 0 0 0 A D

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@arnoutboks #dpc19 Creating the matrix M A D A B C D A 0 1 0 0 B 1 0 1 0 C 0 1 0 1 D 1 1 0 0 Outbound 2 3 1 1 B C

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@arnoutboks #dpc19 Creating the matrix M A D A B C D A 0 1/3 0 0 B 1/2 0 1 0 C 0 1/3 0 1 D 1/2 1/3 0 0 Outbound 2 3 1 1 B C

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@arnoutboks #dpc19 Equation s = Ms A B C D A 0 1/3 0 0 B 1/2 0 1 0 C 0 1/3 0 1 D 1/2 1/3 0 0 sA sB sC sD M s

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@arnoutboks #dpc19 Equation s = Ms A B C D A 0 1/3 0 0 B 1/2 0 1 0 C 0 1/3 0 1 D 1/2 1/3 0 0 sA sB sC sD M s sA sB sC sD

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@arnoutboks #dpc19 Equation s = Ms A B C D A 0 1/3 0 0 B 1/2 0 1 0 C 0 1/3 0 1 D 1/2 1/3 0 0 sA sB sC sD M s sA sB sC sD =

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@arnoutboks #dpc19 Equation s = Ms A B C D A 0 1/3 0 0 B 1/2 0 1 0 C 0 1/3 0 1 D 1/2 1/3 0 0 sA sB sC sD M s sA sB sC sD = “The higher A scores, the more ‘points’ D gets for being one of the two pages A links to”

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@arnoutboks #dpc19 Eigenvalue problem Q: Does there exist a vector s such that for the given matrix M? s = Ms

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@arnoutboks #dpc19 Eigenvalue problem Q: Does there exist a vector s and a number λ such that for the given matrix M? λs = Ms

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@arnoutboks #dpc19 Eigenvalue problem Q: Does there exist a vector s and a number λ such that for the given matrix M? λs = Ms A: Yes, with λ = 1 Fine print: under certain technical conditions. Lookup "Perron–Frobenius theorem” if you’re interested.

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Calculating PageRank Beyond mere existence

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@arnoutboks #dpc19 PageRank as simulated surfing • Someone starts surfing somewhere on the internet • On every page, they follow a random link • What page is shown when the buzzer goes? • Score of a page is the probability of ending up on it

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@arnoutboks #dpc19 PageRank as simulated surfing A B C D A 0 1/3 0 0 B 1/2 0 1 0 C 0 1/3 0 1 D 1/2 1/3 0 0 sA sB sC sD M s sA sB sC sD s = “When on page A, there’s a ½ chance of going to D”

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@arnoutboks #dpc19 Power Method M s(n) s(n+1) transition probabilities probability per page after n clicks probability per page after n+1 clicks

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@arnoutboks #dpc19 Power Method M s(n) s(n+1) transition probabilities probability per page after n clicks probability per page after n+1 clicks s(0) reasonable initial guess

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@arnoutboks #dpc19 Power Method is proven to converge to the eigenvector (for a matrix M like we have)

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@arnoutboks #dpc19 Implementation in PHP aboks/power-iteration (uses markrogoyski/math-php)

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@arnoutboks #dpc19 Implementation in PHP getDominantEigenpair($m); var_dump($pair->getEigenvector());

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@arnoutboks #dpc19 Behind the scenes getM(), 1); $v = new Vector($ones); for ($i = 0; $i < 1000; $i++) { $v = $m->vectorMultiply($v); } return $v; }

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@arnoutboks #dpc19 Implementation concerns Stopping criterion • Number of iterations • Eigenvector tolerance Scaling intermediate results Preventing reducible matrices

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@arnoutboks #dpc19 Preventing reducible matrices A B C D E

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@arnoutboks #dpc19 Damping factor α Google uses α ≈ 0.85

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@arnoutboks #dpc19 Preventing reducible matrices When moving on to a new page: • α probability of following a link • (1- α) probability of ‘teleporting’ to a random page

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@arnoutboks #dpc19 Preventing reducible matrices A B C D A (1/n)(1 – α) (1/n)(1 – α) + 1/3α (1/n)(1 – α) (1/n)(1 – α) B (1/n)(1 – α) + 1/2α (1/n)(1 – α) (1/n)(1 – α) + 1α (1/n)(1 – α) C (1/n)(1 – α) (1/n)(1 – α) + 1/3α (1/n)(1 – α) (1/n)(1 – α) + 1α D (1/n)(1 – α) + 1/2α (1/n)(1 – α) + 1/3α (1/n)(1 – α) (1/n)(1 – α) M’i,j = (1/n)(1 – α) + αMi,j

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Applications of PageRank Beyond web search

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@arnoutboks #dpc19 PageRank in general Analysis of networks/directed graphs: • Edge A → B increases score of B • The higher the score of A, the higher the score of B

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@arnoutboks #dpc19 Influence in social networks B A A follows B

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@arnoutboks #dpc19 Food chains B A A eats B

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@arnoutboks #dpc19 CodeRank function A A calls B function B

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@arnoutboks #dpc19 CodeRank A depends on B class A class B

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@arnoutboks #dpc19 Reverse CodeRank B depends on A class A class B

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@arnoutboks #dpc19 CodeRank for PHP pdepend/pdepend Calculates metrics including CodeRank and Reverse CodeRank

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@arnoutboks #dpc19 Package dependencies B A A depends on B

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@arnoutboks #dpc19 PageRank is not very difficult to implement… …but applications are countless

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@arnoutboks #dpc19 Back to our story… $ git stash pop On branch master Changes to be committed: (use "git reset HEAD ..." to unstage) new file: volleyball-story.txt Dropped refs/stash@{0}(b180f4)

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@arnoutboks #dpc19 Ranking (incomplete) competitions B A A lost to B

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@arnoutboks #dpc19 Ranking (incomplete) competitions B A A lost 2-3 to B 3 2

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@arnoutboks #dpc19 The results # Team Pnt 1 DEO 2 30 2 Netwerk 1 24 3 Punch 6 24 4 Delta 5 23 5 Red Stars 1 23 6 Delta 4 17 7 Kalinko 6 12 8 Kratos 7 7 9 Punch 9 6 10 Sovicos 4 4

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@arnoutboks #dpc19 The results # Team Pnt 1 DEO 2 30 2 Netwerk 1 24 3 Punch 6 24 4 Delta 5 23 5 Red Stars 1 23 6 Delta 4 17 7 Kalinko 6 12 8 Kratos 7 7 9 Punch 9 6 10 Sovicos 4 4 # Team % 1 Punch 6 25.79 2 DEO 2 16.86 3 Red Stars 1 15.31 4 Delta 5 14.13 5 Delta 4 11.03 6 Netwerk 1 8.79 7 Kalinko 6 3.13 8 Kratos '08 7 2.49 9 Punch 9 1.24 10 Sovicos 4 1.23

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@arnoutboks #dpc19 The end

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@arnoutboks #dpc19 Feedback & Questions @arnoutboks @arnoutboks @aboks Arnout Boks Please leave your feedback on joind.in: https://joind.in/talk/fb7e9 We’re hiring!

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@arnoutboks #dpc19 Image Credits • https://www.flickr.com/photos/42283496@N08/4011391702/ • https://pixabay.com/illustrations/google-search-engine-browser- search-76517/ • https://unsplash.com/photos/68ZlATaVYIo • https://imgur.com/gallery/SLHBV • https://www.flickr.com/photos/125329869@N03/14418114781 • https://unsplash.com/photos/ZiQkhI7417A