r - SNAP - Stanford University

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( t+ 1) = Mr ( t) Markov Chains Set of states X Transition matrix P where P ij = P(X t=i | X t-1=j) π specifying the probability of being at each state x ∈ X Goal is to find π such that π = P π 2/7/2012 Jure Leskovec, <strong>Stanford</strong> C246: Mining Massive Datasets 24
Theory of Markov chains Fact: For any start vector, the power method applied to a Markov transition matrix P will converge to a unique positive stationary vector as long as P is stochastic, irreducible and aperiodic. 2/7/2012 Jure Leskovec, <strong>Stanford</strong> C246: Mining Massive Datasets 25
Page 1 and 2: CS246: Mining Massive Datasets Jure
Page 3 and 4: We will cover the following Link An
Page 5 and 6: Each link’s vote is proportional
Page 7 and 8: 3 equations, 3 unknowns, no constan
Page 9 and 10: Suppose page j links to 3 pages, in
Page 11 and 12: y a m ry = ry /2 + ra /2 ra = ry /2
Page 13 and 14: Power Iteration: Set
Page 15 and 16: Where is the surfer at time t+1? F
Page 17 and 18: Example: a b ra 1 0 1 0 = rb 0 1 0
Page 19 and 20: 2 problems: Some pages are “dead
Page 21 and 22: The Google solution for spider trap
Page 23: Teleports: Follow random teleport l
Page 27 and 28: A chain is periodic if there exists
Page 29 and 30: Google’s solution that does it al
Page 31 and 32: a y a = m y 0.8·½+0.2·⅓ 1/3 1/
Page 33 and 34: Construct the N x N matrix A as fol
Page 35 and 36: Key step is matrix-vector multiplic
Page 37 and 38: Encode sparse matrix using only non
Page 39 and 40: Assume enough RAM to fit r new into
Page 41 and 42: Similar to nested-loop join in data
Page 43 and 44: Break M into stripes Each stripe c

r - SNAP - Stanford University

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