The document discusses numerical linear algebra techniques for analyzing large graphs and networks. It provides examples of large social networks like Flickr that can be represented as graphs and analyzed using graph-based algorithms. Specifically, it discusses using techniques like PageRank to analyze link structures and rank nodes in a graph based on their importance. It also discusses computational methods like power iteration and Krylov subspace methods for efficiently solving the large systems of equations that arise in PageRank and related network analysis problems.

1. Numerical Linear Algebra for
Data and Link Analysis
Leonid Zhukov, Yahoo! Inc
David Gleich, Stanford

2. Outline
• Intro
• Scale free graphs
– Flickr example
• Page rank computing

3. Scientific computing
• Engineering Computing • Information Retrieval
– continuum problems, PDE – Discrete data is given, no
governed, control over control over resolution
discretization – No associated physical space
– 2D or 3D (coordinates)
– Uniform distribution of node – Not planar, triangulation
degrees – Power-low degree distribution
– “Small world” effect

4. Large graphs
• Explicit: graphs & networks
– Web graph
– Internet
– Yahoo! Photo Sharing (Flickr)
– Yahoo! 360 (Social network)
• Implicit: transaction history, email, messenger
– Yahoo! Search marketing (Overture)
– Yahoo! Mail
– Yahoo Messenger
• Constructed: affinity between data points
– Yahoo! Music (Launch)
– Yahoo! Movies
– Yahoo! ….

5. Flickr – social network

6. Flickr: adjacency matrix
580,000 users, 3,500,000 links

7. Flickr: reverse Cuthill-McKee
580,000 users, 3,500,000 links

8. Flickr – a graph

9. 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 = 274,649 374 186 155 11
• # connected component = 43,189
• largest connected components = 404,893 378 112 108 103
• highest core number = 249 (size 668)

10. Flickr: scale free graph
in-degrees
out-degrees

11. Data and link analysis
• Web Search
– ranking of pages/hosts/domains
– graph algorithms
– web spam
• Social networks
– finding communities
– trust propagation
• Data mining
– clustering
– classification

12. 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

13. What do we compute?
• Probability matrix:
• Graph Laplacian:
• SVD/LSI:
Computing eigenvalues / eigenvectors, linear systems

14. Yahoo! Search Marketing (Overture)
View Bids Tool

15. 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

16. Overture bid graph
Terms
accessory desk
educational software
Office Max
epson
Buy.com
fax
game eBAY International AG
george harrison Net Business
jackson michael MusicStack
alfred hitchcock Videoflicks.com
Advertisers

17. Overture bid matrix
1 2 3 4 5 6
accessory desk
game 1. Net Business
fax 2. Office Max
educational software 3. Buy.com
epson 4. eBAY
george harrison 5. MusicStack
6. Videoflicks.com
jackson michael
alfred hitchcock

18. 170,077 terms, 25,971 advertisers, 1,307,316 bids

19. Overture bid matrix: 18-core
2000 advertisers , 3000 search terms, 92,347 bids

20. Spectral graph partitioning
• assign each node indicator pi = §1
1 2 3 4 • partition p = {-1,-1,-1,-1, +1, +1, +1}
5
6
7 8
+1
-1
Relaxation procedure:
=>
Quadratic optimization:
Min cut solution :
Rounding off:

21. Spectral: ordering
Eigenvector Eigenvector – sorted
Adjacency matrix Adjacency matrix – re-ordered
perm = [1 2 6 7 3 8 4 5]

22. 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

23. Search Market Place

24. Sample clusters
'A.S.C. Incorporated'
'Atlantic Telecom' 'Berent Associates Center for
'AudioLink' Shyness and Social Therapy'
'Cost Plus Electronics' 'Dr. Puff'
'Headset Express' 'New York Psychotherapy Collective'
'Hello Direct' 'Ruby Shoes'
'PDA Mountain' 'Self-employed psychologist'
'PK TECH INC‘ 'www.kabbalah.com‘
………………….. ……………
'accessory phone' 'health mental'
'cordless headset' 'help self'
'cordless headset phone' 'improvement self'
'free hands' 'parenting‘
'headset'
'headset microphone'
'headset phone'
'headset plantronics'
'headset wireless'
'isdn'
'plantronics‘
…………….

25. Flickr: “10-core” spectral ordering

26. Websearch Engines
• Crawler + indexer • Front end
# Qs PR
1 0.7 0.4
2 0.8 0.1
Link Analysis 3 0.1 0.5
Text Analysis
PageRank Inverted Index

27. 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

28. 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

29. PageRank linear system
• Eigensystem:
• Linear system:

30. Computational diagram
PageRank Problem
Eigensystem Linear system
Power Jacobi
GMRES BiCG BiCGSTAB
Iterations Iterations
Preconditioners
Block Additive
Jacobi
Jacobi Schwarz
PageRank Vector

31. 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

32. Power law graphs ( y = x - b) ?
bs: 0.3M nodes y2: 266 M nodes db: 1B nodes

33. Computational Methods
Error Metrics for Jacobi Convergence Error Metrics for BiCG Convergence
0 2
10 10
||x(k) - x*|| ||x(k) - x*||
-1
(k)
10 ||A x - b|| ||A x(k) - b||
(k) * (k)
||x - x ||/||x || 10
0
||x(k) - x*||/||x(k)||
-2
10
-3
10 -2
Metric Value
Metric Value
10
-4
10
-4
-5 10
10
-6
10
-6
10
-7
10
-8 -8
10 10
0 10 20 30 40 50 60 70 80 0 5 10 15 20 25 30 35 40 45
Iteration Iteration

34. Convergence results
0
uk iteration convergence 0
db iteration convergence
10 10
std std
10
-1 jacobi 10
-1 jacobi
gmres gmres
-2
bicg -2
bicg
10 bcgs 10 bcgs
-3 -3
10 10
Error
Error
-4 -4
10 10
-5 -5
10 10
-6 -6
10 10
-7 -7
10 10
-8
10 -8
0 10 20 30 40 50 60 70 80 10
0 10 20 30 40 50 60 70
Iteration Iteration
uk: 18.5 M pages, 300 M links
db: 70 M pages, 1 B links.

35. Convergence results
0
uk time convergence 0
db time convergence
10 10
std std
10
-1 jacobi -1 jacobi
10
gmres gmres
-2
bicg -2
bicg
10 bcgs 10 bcgs
-3 -3
10 10
Error
Error
-4 -4
10 10
-5 -5
10 10
-6
10 -6
10
-7
10 10
-7
-8
10 10
-8
0 5 10 15 20 25 30 35 40 0 100 200 300 400 500 600 700
Time (sec) Time (sec)
uk: 18.5 M pages, 300 M links db: 70 M pages, 1 B links.

36. Summary
Name Size Power Jacobi GMRES BiCG BCGS
edu 2M 84 84 21* 44* 21*
20 procs 14M 0.09/7.5s 0.07/6.5s 0.6/13.2s 0.4/17.7s 0.4/8.7s
yahoo-r2 14M 71 65 12* 35* 17*
20 procs 266M 1.8/129s 1.9/126s 16/194s 8.6/300s 9.9/168s
uk 18.5M 73 71 22* 25* 11*
60 procs 300M 0.09/7s 0.1/10s 0.8/17.6s 0.8/19.4s 1.0/10.8s
yahoo-r3 60M 76 75
60 procs 850M 1.6/119s 1.5/112s
db 70M 62 58 29 45 15*
60 procs 1B 9.0/557s 8.7/506s 15/432s 15/676s 15/220s
av 1.4B 72 26
140 procs 6.6B 4.6/333s 15/391s

37. Reduced teleportation
2
Convergence and Conditioning for db
10
std
gmres
0
c = 0.85
10 c = 0.90
c = 0.95
c = 0.99
Error/Residual
-2
10
-4
10
-6
10
-8
10
0 20 40 60 80 100 120 140 160 180 200
Iteration
db: 70 M pages, 1 B links.

38. YRL research cluster
Custom
Parallel PageRank
PETSc
Off the shelf
MPI
Gigabit Switch
RLX Blades
Dual 2.8 GHz Xeon
4 GB RAM
Gigabit Ethernet
120 Total
RLX RLX RLX

39. Typical graph
out-degrees.
in-degrees.
bs: 70 M pages, 1 B links.

40. Parallel Matrix-Vector Multiply
=

41. Graph distribution methods
• Balance rows
• Balance nnz
• Random destribution
• Balance rows and nnz (heuristics)
• 2D distributions
• Graph partitioning
• Hypergraph partitioning

42. Some parallel experiments

43. Domain lexicographic sorting
http://host.domain.tld/path ! http://tld.domain.host/path
bs-cc: 17 k pages, 133 k links

44. Domain ordering
0
Host Order Improvement
10
std-140
10
-1 bcgs-140
std-140-host
-2 bcgs-140-host
10
-3
10
error/residual
-4
10
-5
10
-6
10
-7
10
-8
10
0 100 200 300 400 500 600 700 800 900
Time (sec)
av: 1.4 B pages, 6.6 B links.

45. Full web graph parallelization
Scaling for computing with full-web
Performance Increase (Percent decrease in time)
250%
std
bcgs
200%
150%
100%
50%
0%
90% 100% 110% 120% 130% 140% 150% 160% 170%
140 186 226
av: 1.4 B pages, 6.6 B links.

46. FEM mesh for CFD computations

47. Power-law graph embedding in 3D

48. final thoughts
• Power-law graphs are different
• Not all power-law graphs are the same
• But all have central core, chains, singletons.
• Difficult to distribute / parallelize
• Its is experimental data
• Lots of cool applications
• Let’s talk …..

49. 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 (2002-03), A. Langville (2002), G. Jeh
(2003), S. Kamvar (2003), A. Broder (2004)