Numerical Linear Algebra for Data and Link Analysis
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1 Numerical Linear Algebra for Data and Link Analysis Leonid Zhukov, Yahoo! Inc David Gleich, Stanford SIAM PP-2006, San Francisco, CA
2 Scientific computing Engineering Computing Information Retrieval continuum problems, PDE governed, control over discretization 2D or 3D Uniform distribution of node degrees Discrete data is given, no control over resolution No associated physical space (coordinates) Not planar, triangulation Power-low degree distribution Small world effect
3 Large graphs Explicit: graphs & networks Web graph Internet Yahoo! Photo Sharing (Flickr) Yahoo! 360 (Social network) Implicit: transaction history, , messenger Yahoo! Search marketing (Overture) Yahoo! Mail Yahoo Messenger Constructed: affinity between data points Yahoo! Music (Launch) Yahoo! Movies Yahoo!.
4 Flickr social network
5 Flickr: adjacency matrix 580,000 users, 3,500,000 links
6 Flickr: reverse Cuthill-McKee 580,000 users, 3,500,000 links
7 Flickr a graph
8 Flickr: some stats # nodes = 584,207 # edges (nnz) = 3,555,115 max in-degree = 3531 max out-degree = 8976 < in-degree > = < out-degree > = 6 diameter = 18 # strongly connected components = 152,324 largest strongly connected components = 274, # connected component = 43,189 largest connected components = 404, highest core number = 249 (size 668)
9 Flickr: scale free graph in-degrees out-degrees
10 Data and links analysis Graph partitioning: balanced, for data distribution natural boundaries, for clustering / communities Numerical computing on graphs: Node ranking methods (PageRank) Dimensionality reduction (SVD, HITS) Low dimensional embeddings Graph interpolation / regularization
11 What do we compute? Probability matrix: Graph Laplacian: SVD/LSI: Computing eigenvalues / eigenvectors, linear systems
12 Yahoo! Search Marketing (Overture) View Bids Tool
13 Overture data search $ bid advertiser accessory desk.83 Office Max alfred hitchcock.01 Videoflicks.com educational software.13 Buy.com educational software.4 ebay International AG educational software.17 OfficeMax epson.28 Buy.com epson.4 ebay International AG fax.13 Buy.com fax.4 ebay International AG fax.38 OfficeMax game.02 Net Business game.25 Buy.com george harrison.15 ebay International AG george harrison.05 MusicStack george harrison.01 Videoflicks.com jackson michael.05 MusicStack jackson michael.01 Videoflicks.com
14 Overture bid graph Terms accessory desk educational software epson fax game george harrison jackson michael alfred hitchcock Office Max Buy.com ebay International AG Net Business MusicStack Videoflicks.com Advertisers
15 Overture bid matrix accessory desk game fax educational software epson george harrison jackson michael 1. Net Business 2. Office Max 3. Buy.com 4. ebay 5. MusicStack 6. Videoflicks.com alfred hitchcock
16 170,077 terms, 25,971 advertisers, 1,307,316 bids
17 Overture bid matrix: 18-core 2000 advertisers, 3000 search terms, 92,347 bids
18 Spectral graph partitioning assign each node indicator p i = ±1 partition p = {-1,-1,-1,-1, +1, +1, +1} Relaxation procedure: => Quadratic optimization: Min cut solution : Rounding off:
19 Spectral: ordering Eigenvector Eigenvector sorted Adjacency matrix Adjacency matrix re-ordered perm = [ ]
20 Spectral : additions Recursively Min cut, ratio cut, normalized cut, quotient cut, Sweep for optimal value Flow based refinement Optimize partition among several eigenvectors Bi-partite graphs + other bells & whistles
21 Search Market Place
22 Search Market Place
23 Search Market Place
24 Search Market Place
25 Sample clusters 'A.S.C. Incorporated' 'Atlantic Telecom' 'AudioLink' 'Cost Plus Electronics' 'Headset Express' 'Hello Direct' 'PDA Mountain' 'PK TECH INC.. 'accessory phone' 'cordless headset' 'cordless headset phone' 'free hands' 'headset' 'headset microphone' 'headset phone' 'headset plantronics' 'headset wireless' 'isdn' 'plantronics. 'Berent Associates Center for Shyness and Social Therapy' 'Dr. Puff' 'New York Psychotherapy Collective' 'Ruby Shoes' 'Self-employed psychologist' ' 'health mental' 'help self' 'improvement self' 'parenting
26 Flickr: 10-core spectral ordering
27 Flickr: 10-core spectral ordering
28 Websearch Engines Crawler + indexer Front end Link Analysis PageRank Text Analysis Inverted Index # Qs PR
29 PageRank model Random walk on graph Markov process: memory-less, homogeneous Stationary distribution: existence, uniqueness, convergence Perron-Frobenius theorem, irreducible, every state is reachable from every other, and aperiodic no cycles
30 PageRank Construct probability matrix: Construct transition row-stochastic matrix for Markov process Correct reducibility: Find stationary distribution ( exist & unique): c teleportation coefficient, v personalization vector, d dangling nodes (sinks) indicator
31 PageRank linear system Eigensystem: Linear system:
32 Computational diagram PageRank Problem Eigensystem Linear system Power Iterations Jacobi Iterations GMRES BiCG BiCGSTAB Preconditioners Jacobi Block Jacobi Additive Schwarz PageRank Vector
33 Data: WebGraphs Name # Nodes/size # Links / nnz Memory bs 0.3M 1.6M 20.4 MB edu 2M 14M 176MB yahoo-r2 14M 266M 3.25GB uk 18.5M 300M 3.67GB yahoo-r3 60M 850M 10.4GB db 70M 1B 12.3GB av 1.4B 6.6B 80GB Graph in memory, compressed storage, int, int, double
34 Power law graphs ( y = x -b )? bs: 0.3M nodes y2: 266 M nodes db: 1B nodes
35 Computational Methods 10 0 Error Metrics for Jacobi Convergence x (k) - x * A x (k) - b x (k) - x * / x (k) 10 2 Error Metrics for BiCG Convergence x (k) - x * A x (k) - b 10 0 x (k) - x * / x (k) Metric Value Metric Value Iteration Iteration
36 Convergence results uk iteration convergence std jacobi gmres bicg bcgs db iteration convergence std jacobi gmres bicg bcgs Error 10-4 Error Iteration uk: 18.5 M pages, 300 M links Iteration db: 70 M pages, 1 B links.
37 Convergence results uk time convergence std jacobi gmres bicg bcgs db time convergence std jacobi gmres bicg bcgs Error 10-4 Error Time (sec) uk: 18.5 M pages, 300 M links Time (sec) db: 70 M pages, 1 B links.
38 Summary Name Size Power Jacobi GMRES BiCG BCGS edu 20 procs yahoo-r2 20 procs uk 60 procs yahoo-r3 60 procs db 60 procs av 140 procs 2M 14M 14M 266M 18.5M 300M 60M 850M 70M 1B 1.4B 6.6B /7.5s /129s /7s /119s /557s /333s /6.5s /126s /10s /112s /506s 21* 0.6/13.2s 12* 16/194s 22* 0.8/17.6s 29 15/432s 44* 0.4/17.7s 35* 8.6/300s 25* 0.8/19.4s 45 15/676s 21* 0.4/8.7s 17* 9.9/168s 11* 1.0/10.8s 15* 15/220s 26 15/391s
39 Reduced teleportation Error/Residual Convergence and Conditioning for db std gmres c = 0.85 c = 0.90 c = 0.95 c = db: 70 M pages, 1 B links. Iteration
40 YRL research cluster Parallel PageRank MPI PETSc Custom Off the shelf Gigabit Switch RLX RLX RLX RLX Blades Dual 2.8 GHz Xeon 4 GB RAM Gigabit Ethernet 120 Total
41 Typical graph out-degrees. in-degrees. bs: 70 M pages, 1 B links.
42 Parallel Matrix-Vector Multiply =
43 Graph distribution methods Balance rows Balance nnz Random destribution Balance rows and nnz (heuristics) 2D distributions Graph partitioning Hypergraph partitioning
44 Some parallel experiments
45 Domain ordering Host Order Improvement std-140 bcgs-140 std-140-host bcgs-140-host error/residual av: 1.4 B pages, 6.6 B links. Time (sec)
46 Full web graph parallelization Performance Increase (Percent decrease in time) 250% 200% 150% 100% 50% std bcgs Scaling for computing with full-web 0% 90% 100% 110% 120% 130% 140% 150% 160% 170% av: 1.4 B pages, 6.6 B links.
47 FEM mesh for CFD computations
48 Power-law graph embedding in 3D
49 Final thoughts Power-law graphs are different Have central core, long chains, clusters, singletons Not all power-law graphs are the same Difficult to distribute / parallelize Noise: its is experimental data after all Lots of cool applications Let s talk..
50 References Spectral graph partitioning: - M. Fiedler (1973), A. Pothen (1990), H. Simon (1991), B. Mohar (1992), B. Hendrickson (1995), D. Spielman (1996), F. Chang (1996), S. Guattery (1998), R. Kannan (1999), J. Shi (2000), I. Dhillon ( 2001), A. Ng (2001), H. Zha (2001), C. Ding (2001) PageRank computing: - S.Brin (1998), L. Page (1998), J. Kleinberg (1999), A. Arasu (2002), T. Haveliwala ( ), A. Langville (2002), G. Jeh (2003), S. Kamvar (2003), A. Broder (2004)
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