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Ms y 0 a 0 1 ms yahoo a 0 0 r mr amazon msoft y

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Unformatted text preview: g Massive Datasets y ½½0 a =½ 0 1 m 0½0 y a m 21 Simple iterative scheme Suppose there are N web pages Initialize: r0 = [1/N,….,1/N]T Iterate: rk+1 = M∙rk Stop when |rk+1 - rk|1 < ε |x|1 = ∑1≤i≤N|xi| is the L1 norm Can use any other vector norm e.g., Euclidean 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 22 Power iteration: A MS MS Y! Y! A ½ ½ 0 A Set ri=1/n ri=∑j Mij∙rj And iterate Y! ½ 0 1 MS 0 ½ 0 Example: y a= m 2/7/2011 1/3 1/3 1/3 1/3 1/2 1/6 5/12 1/3 1/4 3/8 11/24 . . . 1/6 Jure Leskovec, Stanford C246: Mining Massive Datasets 2/5 2/5 1/5 23 Imagine a random web surfer At any time t, surfer is on some page P At time t+1, the surfer follows an outlink from P uniformly at random Ends up on some page Q linked from P Process repeats indefinitely Let p(t) be a vector whose ith component is the probability that the surfer is at page i at time t p(t) is a probability distribution on pages 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 24 Where is the surfer at time t+1? Follows a link uniformly at random p(t+1) = M∙p(t) Suppose the random walk reaches a state such that p(t+1) = M∙p(t) = p(t) Then p(t) is called a stationary distribution for the random walk Our rank vector r satisfies r = M∙r So it is a stationary distribution for the random surfer 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 25 A central result from the theory of random walks (aka Markov processes): For graphs that satisfy certain conditions, the stationary distribution is unique and eventually will be reached no matter what the initial probability distribution at time t = 0. 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 26 Some pages are “dead ends” (have no out-links) Such pages cause importance to leak out Spider traps (all out links a...
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