Random Walks2-D ArraysPageRank90-10 RuleWeb Graph Input FormatTransition MatrixWeb Graph to Transition MatrixMonte Carlo SimulationRandom SurferSlide 10Random Surfer: Monte Carlo SimulationMathematical ContextThe Power MethodSlide 14Slide 15Slide 16PowerPoint PresentationRandom Surfer: Scientific ChallengesCompSci 100E3.1Random Walks•“A drunk man wil l find his way home, but a drunk bird may get lost forever” – Shizuo Kakutani •Suppose you proceed randomly from the middle of a large city whose streets form a completegridWill you make it back home?Will you escape?•What if you mark the intersections you’ve visited?Self avoiding random walkApplications to chemistry: polymer formationCompSci 100E3.22-D Arrays•Want to arrange information in rows and columns•Initializedouble[][] a = new double[M][N];•Nested loops to accessfor (int i = 0; i < M; i+= 1) for (int j = 0; j < N; j+= 1) a[i][j] = 0; •Represent as 1-D array?CompSci 100E3.3PageRank•Google's PageRank™ algorithm. [Sergey Brin and Larry Page, 1998]Measure popularity of pages based on hyperlink structure of Web.Revolutionized access to world's information.CompSci 100E3.490-10 Rule•Model. Web surfer chooses next page:90% of the time surfer clicks random hyperlink.10% of the time surfer types a random page.•Caveat. Crude, but useful, web surfing model.No one chooses links with equal probability.No real potential to surf directly to each page on the web.The 90-10 breakdown is just a guess.It does not take the back button or bookmarks into account.We can only afford to work with a small sample of the web.…CompSci 100E3.5Web Graph Input Format•Input format.N pages numbered 0 through N-1.Represent each hyperlink with a pair of integers.Graph representationCompSci 100E3.6Transition Matrix•Transition matrix. p[i][j]= prob. that surfer moves from page i to j.surfer on page 1 goes topage 2 next 38% of the timeCompSci 100E3.7Web Graph to Transition Matrix% java Transition tiny.txt5 5 0.02000 0.92000 0.02000 0.02000 0.02000 0.02000 0.02000 0.38000 0.38000 0.20000 0.02000 0.02000 0.02000 0.92000 0.02000 0.92000 0.02000 0.02000 0.02000 0.02000 0.47000 0.02000 0.47000 0.02000 0.02000CompSci 100E3.8Monte Carlo Simulation•Monte Carlo simulation.Surfer starts on page 0.Repeatedly choose next page, according to transition matrix.Calculate how often surfer visits each page.transition matrixpageHow? see next slideCompSci 100E3.9Random Surfer•Random move. Surfer is on page page. How to choose next page j?Row page of transition matrix gives probabilities.Compute cumulative probabilities for row page. Generate random number r between 0.0 and 1.0.Choose page j corresponding to interval where r lies.pagetransition matrixCompSci 100E3.10Random Surfer•Random move. Surfer is on page page. How to choose next page j?Row page of transition matrix gives probabilities.Compute cumulative probabilities for row page. Generate random number r between 0.0 and 1.0.Choose page j corresponding to interval where r lies.// make one random movedouble r = Math.random();double sum = 0.0;for (int j = 0; j < N; j++) { // find interval containing r sum += p[page][j]; if (r < sum) { page = j; break; }}CompSci 100E3.11Random Surfer: Monte Carlo Simulationpublic class RandomSurfer { public static void main(String[] args) { int T = Integer.parseInt(args[0]); // number of moves int N = in.nextInt(); // number of pages int page = 0; // current page double[][] p = new int[N][N]; // transition matrix // read in transition matrix ... // simulate random surfer and count page frequencies int[] freq = new int[N]; for (int t = 0; t < T; t++) { // make one random move freq[page]++; } // print page ranks for (int i = 0; i < N; i++) { System.out.println(String.format("%8.5f", (double) freq[i] / T); } System.out.println(); }}see previous slidepage rankCompSci 100E3.12Mathematical Context•Convergence. For the random surfer model, the fraction of time the surfer spends on each page converges to a unique distribution, independent of the starting page.€ 428,6711,570,055, 417,2051,570,055, 229,5191,570,055, 388,1621,570,055, 106,4981,570,055 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥"page rank""stationary distribution" of Markov chain"principal eigenvector" of transition matrixCompSci 100E3.13The Power Method•Q. If the surfer starts on page 0, what is the probability that surfer ends up on page i after one step? •A. First row of transition matrix.CompSci 100E3.14The Power Method•Q. If the surfer starts on page 0, what is the probability that surfer ends up on page i after two steps?•A. Matrix-vector multiplication.CompSci 100E3.15The Power Method•Power method. Repeat until page ranks converge.CompSci 100E3.16Mathematical Context•Convergence. For the random surfer model, the power method iterates converge to a unique distribution, independent of the starting page."page rank""stationary distribution" of Markov chain"principal eigenvector" of transition matrixCompSci 100E3.17CompSci 100E3.18Random Surfer: Scientific Challenges•Google's PageRank™ algorithm. [Sergey Brin and Larry Page, 1998]Rank importance of pages based on hyperlink structure of web,using 90-10 rule.Revolutionized access to world's information.•Scientific challenges. Cope with 4 billion-by-4 billion matrix!Need data structures to enable computation.Need linear algebra to fully understand
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