1 / 37

CS345 Data Mining

CS345 Data Mining. Link Analysis Algorithms Page Rank. Anand Rajaraman, Jeffrey D. Ullman. Link Analysis Algorithms. Page Rank Hubs and Authorities Topic-Specific Page Rank Spam Detection Algorithms Other interesting topics we won’t cover Detecting duplicates and mirrors

basil
Download Presentation

CS345 Data Mining

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CS345Data Mining Link Analysis Algorithms Page Rank Anand Rajaraman, Jeffrey D. Ullman

  2. Link Analysis Algorithms • Page Rank • Hubs and Authorities • Topic-Specific Page Rank • Spam Detection Algorithms • Other interesting topics we won’t cover • Detecting duplicates and mirrors • Mining for communities

  3. Ranking web pages • Web pages are not equally “important” • www.joe-schmoe.com v www.stanford.edu • Inlinks as votes • www.stanford.edu has 23,400 inlinks • www.joe-schmoe.com has 1 inlink • Are all inlinks equal? • Recursive question!

  4. Simple recursive formulation • Each link’s vote is proportional to the importance of its source page • If page P with importance x has n outlinks, each link gets x/n votes • Page P’s own importance is the sum of the votes on its inlinks

  5. Yahoo Amazon M’soft Simple “flow” model The web in 1839 y = y /2 + a /2 a = y /2 + m m = a /2 y/2 y a/2 y/2 m a/2 m a

  6. Solving the flow equations • 3 equations, 3 unknowns, no constants • No unique solution • All solutions equivalent modulo scale factor • Additional constraint forces uniqueness • y+a+m = 1 • y = 2/5, a = 2/5, m = 1/5 • Gaussian elimination method works for small examples, but we need a better method for large graphs

  7. Matrix formulation • Matrix M has one row and one column for each web page • Suppose page j has n outlinks • If j ! i, then Mij=1/n • Else Mij=0 • M is a columnstochastic matrix • Columns sum to 1 • Suppose r is a vector with one entry per web page • ri is the importance score of page i • Call it the rank vector • |r| = 1

  8. j i i = 1/3 M r Example Suppose page j links to 3 pages, including i r

  9. Eigenvector formulation • The flow equations can be written r = Mr • So the rank vector is an eigenvector of the stochastic web matrix • In fact, its first or principal eigenvector, with corresponding eigenvalue 1

  10. Yahoo r = Mr Amazon M’soft y 1/2 1/2 0 y a = 1/2 0 1 a m 0 1/2 0 m Example y a m y 1/2 1/2 0 a 1/2 0 1 m 0 1/2 0 y = y /2 + a /2 a = y /2 + m m = a /2

  11. Power Iteration method • Simple iterative scheme (aka relaxation) • Suppose there are N web pages • Initialize: r0 = [1/N,….,1/N]T • Iterate: rk+1 = Mrk • Stop when |rk+1 - rk|1 <  • |x|1 = 1≤i≤N|xi| is the L1 norm • Can use any other vector norm e.g., Euclidean

  12. Yahoo Amazon M’soft Power Iteration Example y a m y 1/2 1/2 0 a 1/2 0 1 m 0 1/2 0 y a = m 1/3 1/3 1/3 1/3 1/2 1/6 5/12 1/3 1/4 3/8 11/24 1/6 2/5 2/5 1/5 . . .

  13. Random Walk Interpretation • 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

  14. The stationary distribution • Where is the surfer at time t+1? • Follows a link uniformly at random • p(t+1) = Mp(t) • Suppose the random walk reaches a state such that p(t+1) = Mp(t) = p(t) • Then p(t) is called a stationary distribution for the random walk • Our rank vector r satisfies r = Mr • So it is a stationary distribution for the random surfer

  15. Existence and Uniqueness 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.

  16. Spider traps • A group of pages is a spider trap if there are no links from within the group to outside the group • Random surfer gets trapped • Spider traps violate the conditions needed for the random walk theorem

  17. Microsoft becomes a spider trap Yahoo y a m y 1/2 1/2 0 a 1/2 0 0 m 0 1/2 1 Amazon M’soft y a = m 1 1 1 1 1/2 3/2 3/4 1/2 7/4 5/8 3/8 2 0 0 3 . . .

  18. Random teleports • The Google solution for spider traps • At each time step, the random surfer has two options: • With probability , follow a link at random • With probability 1-, jump to some page uniformly at random • Common values for  are in the range 0.8 to 0.9 • Surfer will teleport out of spider trap within a few time steps

  19. y y y 1/3 1/3 1/3 y 1/2 a 1/2 m 0 1/2 1/2 0 + 0.2* 0.8* Random teleports ( = 0.8) 0.2*1/3 1/2 Yahoo 0.8*1/2 1/2 0.2*1/3 0.8*1/2 0.2*1/3 1/2 1/2 0 1/2 0 0 0 1/2 1 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 + 0.2 Amazon M’soft 0.8 y 7/15 7/15 1/15 a 7/15 1/15 1/15 m 1/15 7/15 13/15

  20. Random teleports ( = 0.8) 1/2 1/2 0 1/2 0 0 0 1/2 1 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 + 0.2 Yahoo 0.8 y 7/15 7/15 1/15 a 7/15 1/15 1/15 m 1/15 7/15 13/15 Amazon M’soft y a = m 1 1 1 1.00 0.60 1.40 0.84 0.60 1.56 0.776 0.536 1.688 7/11 5/11 21/11 . . .

  21. Matrix formulation • Suppose there are N pages • Consider a page j, with set of outlinks O(j) • We have Mij = 1/|O(j)| when j!i and Mij = 0 otherwise • The random teleport is equivalent to • adding a teleport link from j to every other page with probability (1-)/N • reducing the probability of following each outlink from 1/|O(j)| to /|O(j)| • Equivalent: tax each page a fraction (1-) of its score and redistribute evenly

  22. Page Rank • Construct the N£N matrix A as follows • Aij = Mij + (1-)/N • Verify that A is a stochastic matrix • The page rank vectorr is the principal eigenvector of this matrix • satisfying r = Ar • Equivalently, r is the stationary distribution of the random walk with teleports

  23. Dead ends • Pages with no outlinks are “dead ends” for the random surfer • Nowhere to go on next step

  24. Non- stochastic! Microsoft becomes a dead end 1/2 1/2 0 1/2 0 0 0 1/2 0 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 + 0.2 Yahoo 0.8 y 7/15 7/15 1/15 a 7/15 1/15 1/15 m 1/15 7/15 1/15 Amazon M’soft y a = m 1 1 1 1 0.6 0.6 0.787 0.547 0.387 0.648 0.430 0.333 0 0 0 . . .

  25. Dealing with dead-ends • Teleport • Follow random teleport links with probability 1.0 from dead-ends • Adjust matrix accordingly • Prune and propagate • Preprocess the graph to eliminate dead-ends • Might require multiple passes • Compute page rank on reduced graph • Approximate values for deadends by propagating values from reduced graph

  26. Computing page rank • Key step is matrix-vector multiplication • rnew = Arold • Easy if we have enough main memory to hold A, rold, rnew • Say N = 1 billion pages • We need 4 bytes for each entry (say) • 2 billion entries for vectors, approx 8GB • Matrix A has N2 entries • 1018 is a large number!

  27. Rearranging the equation r = Ar, where Aij = Mij + (1-)/N ri = 1≤j≤N Aij rj ri = 1≤j≤N [Mij + (1-)/N] rj =  1≤j≤N Mij rj + (1-)/N 1≤j≤N rj =  1≤j≤N Mij rj + (1-)/N, since |r| = 1 r = Mr + [(1-)/N]N where [x]N is an N-vector with all entries x

  28. Sparse matrix formulation • We can rearrange the page rank equation: • r = Mr + [(1-)/N]N • [(1-)/N]N is an N-vector with all entries (1-)/N • M is a sparse matrix! • 10 links per node, approx 10N entries • So in each iteration, we need to: • Compute rnew = Mrold • Add a constant value (1-)/N to each entry in rnew

  29. Sparse matrix encoding • Encode sparse matrix using only nonzero entries • Space proportional roughly to number of links • say 10N, or 4*10*1 billion = 40GB • still won’t fit in memory, but will fit on disk source node degree destination nodes

  30. Basic Algorithm • Assume we have enough RAM to fit rnew, plus some working memory • Store rold and matrix M on disk Basic Algorithm: • Initialize: rold = [1/N]N • Iterate: • Update: Perform a sequential scan of M and rold to update rnew • Write out rnew to disk as rold for next iteration • Every few iterations, compute |rnew-rold| and stop if it is below threshold • Need to read in both vectors into memory

  31. Update step Initialize all entries of rnew to (1-)/N For each page p (out-degree n): Read into memory: p, n, dest1,…,destn, rold(p) for j = 1..n: rnew(destj) += *rold(p)/n rold rnew src degree destination 0 0 1 1 2 2 3 3 4 4 5 5 6 6

  32. Analysis • In each iteration, we have to: • Read rold and M • Write rnew back to disk • IO Cost = 2|r| + |M| • What if we had enough memory to fit both rnew and rold? • What if we could not even fit rnew in memory? • 10 billion pages

  33. Block-based update algorithm rold rnew src degree destination 0 0 1 1 2 3 2 4 3 5 4 5

  34. Analysis of Block Update • Similar to nested-loop join in databases • Break rnew into k blocks that fit in memory • Scan M and rold once for each block • k scans of M and rold • k(|M| + |r|) + |r| = k|M| + (k+1)|r| • Can we do better? • Hint: M is much bigger than r (approx 10-20x), so we must avoid reading it k times per iteration

  35. Block-Stripe Update algorithm src degree destination rnew 0 rold 1 0 1 2 3 2 4 3 5 4 5

  36. Block-Stripe Analysis • Break M into stripes • Each stripe contains only destination nodes in the corresponding block of rnew • Some additional overhead per stripe • But usually worth it • Cost per iteration • |M|(1+) + (k+1)|r|

  37. Next • Topic-Specific Page Rank • Hubs and Authorities • Spam Detection

More Related