Exploiting Web Matrix Permutations to Speedup PageRank Computation

1. Exploiting Web Matrix Permutations to Speedup PageRank Computation Presented by: Aries Chan, Cody Lawson, and Michael Dwyer

2. Introduction • Internet Statistics • 151 million active internet user as of January 2004 • 76% used a search engine at least once during the month • Average time spent searching was about 40 minutes

3. Introduction • Search Engines • Most common means of accessing Web • Easiest method of organizing and accessing information • Therefore, high quality / usable search engines are very important

4. Introduction • Basic Idea of a Search Engine • Scanning Web Graph • Using crawlers to create an index of the information • Ranking • Organizing and ordering information in a usable way

5. Introduction • Webpage Ranking Problems • Personalization • http://www.google.com/history/?hl=en • Updating – Keeping order up to date • 25% of links are changed in one week • 5% “new content” in one week • 35% of entire web changed in eleven weeks • Reducing Computation Time

6. Introduction • Accelerating the PageRank • Use compression to fit the Web Graph into main memory • Use sequence of scans of the Web Graph to efficiently compute in external memory • Reduce computation time through numerical methods

7. Introduction • Reduce computation time through numerical methods • Use iterative methods such as Gauss-Seidel and Jacobi method (Arasu et al and Bianchini et al) • Subtract off estimates of non-principal eigenvectors (Kamvar et al ) • Sort the graph lexicographically by url resulting in an approximate block structure (Kamvar et al) • Split into two problems: dangling nodes and non-dangling nodes (Lee et al) • Others have been working on ways to only update the ranks of nodes influenced by Web Graph changes

8. Introduction • Del Corso, Gulli, and Romani(authors of the paper) • Numerical optimization of PageRank • View the PageRank computation as a linear system • Transform a dense problem into one which uses a sparse matrix • Treatment of dangling nodes which naturally adapts to the random surfer model • Exploiting web matrix permutations • Increase data locality and reduce number of iterations necessary to solve the problem

9. Google’s PageRank Overview • Web as an oriented graph • The random surfer • vi = ∑ pji vj (sum of PageRanks of nodes linking to i weighted by the transition probability) • Equilibrium distribution • vT = vTH (left eigenvector of H with eigenvalue 1)

10. Problems with the ideal model • Dangling nodes (trap the user) • Impose a random jump to every other page • B = H + auT • Cyclic paths in the Web Graph (reducibility) • Brin and Page suggested adding artificial transitions (low probability jump to all nodes) • G = B + (1  )euT

11. Current PageRank Solution • Since G is just a rank one modification of H, the power method takes advantage of the sparsity of matrix H.

12. Google’s PageRank eigenproblem as a linear system • Expand • vTG = vT(eigenproblem) • G = (H + auT) + (1  )euT (expansion of Google matrix) •  vT(H + auT) + (1  )vTeuT = vT • Restructure • Taking S = I HTuaT • And with vTe = 1 •  Sv = (1 )u (after taking transpose and rearranging)

13. Dangling Nodes Problems • Dangling Nodes • Pages with no links to other pages • Pages whose existence is inferred but crawler has not reached • According to a 2001 sample, approximately 76% are dangling nodes.

14. Dangling Nodes Problems • NaturalModel • B = H + auT • Jump with probability 1 to any other node • Drastic Model • Completely removes dangling nodes • Problems • Dangling nodes themselves are not ranked • Removing nodes create new dangling nodes • Self-loop Model eg. • B = H + F • Fij = {1 if i = j & dangling; 0 otherwise} • Still row stochastic and is similar to natural • Problem • Gives unreasonably high rank to the dangling nodes

15. Which model is the best? • Natural model • the most “accurate”. • Problem • Gives a much more dense matrix B • It is at least partially for this reason that the problem is approached as an eigenproblem to exploit the sparsity of H • Can we have an equally lightweight iterative approach?

16. Iterative Approach with Sparsity • Sparse matrix R • R = I HT • PageRankv obtained by solving • Ry = u, v = y such that ||v||1=1 • Why does this work? • Since S = R uaT • andSv = (1 )u • (R uaT)v = (1 )u • Use Sherman-Morrison formula to calculate the inverse

17. Iterative Approach with Sparsity • We get • The vector v is our PageRank vector and was solved using sparse matrix R

18. Exploiting Web Matrix Permutations • Use a variety of “cheap” operators to permute the web graph in an organized way in hopes to: • increase data locality • reduce the number of iterations in order to solve the problem • Explore different iterative methods in order to solve the problem quicker. • Compare the performance of different iterative methods based on specifically permuted web graphs.

19. Permutation Strategies • The following operations were chosen based on their “limited impact on the computational cost” (Del Corso et al.) • O- orders nodes by increasing out-degree • Q- orders nodes by decreasing out-degree • X- orders nodes by increasing in-degree • Y- orders nodes by decreasing in-degree • B- ordering the nodes according to their BFS (Breadth First Search) order of visit • T- transposes the matrix

20. Permutation Strategies (cont.) • O, Q, X, Y, and T operators  a full matrix • The B operator conveniently arranges R into a lower block triangular structure due to BFS order of visit. • Combining these operations on R, the following structures of the permuted web matrix are produced.

21. Permutation Strategies (cont.) • Visual representations of the permuted web graph.

22. Iterative Strategies • Power Method • Computes dominant eigenvector • Jacobi Method • Using an initial guess, approximates the solution to the linear system of equations. Each successive iteration uses the previous approximated solution as its next guess till a degree of convergence is reached. (*further explained) • Gauss-Seidel Method (Reverse Gauss-Seidel) • Modification of the Jacobi Method, which approximates the solution to each successive equation in the linear system based on the values derived from the previous equations, all within each iterative loop. (*further explained) *(http://college.cengage.com/mathematics/larson/elementary_linear/5e/students/ch08-10/chap_10_2.pdf)

23. Iterative Strategies (cont.) • Further exploration led to iterative methods based on the distinct block structure of certain web graph permutations. • DN (or DNR) Method • The permuted matrix ROT has the property that the lower diagonal block coincides with the identity matrix. The matrix can be easily partitioned into non-dangling and dangling portions. Then the non-dangling part is solved by Gauss-Seidel (or Reverse Gauss-Seidel respectively). • LB/UB/LBR/UBR Methods • Uses the Gauss-Seidel or Reverse Gauss-Seidel methods to solve the individual blocks of the triangular block matrices produced by the B operator.

24. Results

25. Results (cont.) • For both the Power Method and Jacobi Method, the number of Mflops is not dependent on permutations of the web matrix. (“the small differences in numerical data are due to the finite precision” [Del Corso et al.]) • The Jacobi Method (applied to the matrix R) is only a slight improvement (about 3%) compared to the Power Method (applied to the matrix G).

26. Results (cont.) • The Gauss-Seidel and Reverse Gauss-Seidel Methods reduced Mflops by around 40% and running time by around 45% on the full matrix compared to the Power Method on the full matrix. • In particular the Reverse Gauss-Seidel Method performed on the permuted matrix RYTB reduced the number of Mflops by 51% and running time by 82% when compared to the Power method on the full matrix.

27. Results (cont.) • The block methods achieved even better results • In particular, the best overall reduction in computation time and number of Mflops was achieved by LBR method on the permuted matrix RQTB. 58% reduction in terms of Mflops 89% reduction in terms of running time

28. Conclusion • Objective: • Accelerating Google PageRank by numerical methods • Contribution: • Viewed web matrix as a sparse linear system • Formalized new method for treating dangling nodes • Explored new iterative methods and applied them to web matrix permutations • Achievement: • 1/10 of the computation time • Reduced over 50% Mflops

29. References • G.M. Del Corso, A. Gulli, F. Romani, “Exploiting Web Matrix Permutations to Speedup PageRank Computation” • Nielsen MegaView Search (http://en-us.nielsen.com/rankings/insights/rankings/internet)