1. FAST KATZ AND COMMUTERS
Quadrature Rules and Sparse Linear Solvers
for Link Prediction Heuristics
David F. Gleich
Sandia National Labs
Berkeley LAPACK Seminar
February 3, 2011
With Pooya Esfandiar, Francesco Bonchi, Chen Grief,
Laks V. S. Lakshmanan, and Byung-Won On
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin
Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
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2. MAIN RESULTS
A – adjacency matrix For Katz
L – Laplacian matrix Compute one fast
Compute top fast
Katz score :
Commute time:
For Commute
Compute one fast
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3. OUTLINE
Why study these measures?
Katz Rank and Commute Time
How else do people compute them?
Quadrature rules for pairwise scores
Sparse linear systems solves for top-k
As many results as we have time for…
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4. WHY? LINK PREDICTION
Neighborhood based
Path based
Liben-Nowell and Kleinberg 2003, 2006 found that path based link prediction was more efficient
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5. NOTE
All graphs are undirected
All graphs are connected
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7. NOT QUITE, WIKIPEDIA
: adjacency, : random walk
PageRank
Katz
These are equivalent if has constant
degree
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8. WHAT KATZ ACTUALLY SAID
“we assume that each link independently has the
same probability of being effective” …
“we conceive a constant , depending
on the group and the context of the particular
investigation, which has the force of a probability
of effectiveness of a single link. A k-step chain
then, has probability of being effective.”
“We wish to find the column sums of the matrix”
Leo Katz 1953, A New Status Index Derived from Sociometric Analysis, Psychometria 18(1):39-43
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9. A MODERN TAKE
The Katz score (node-based) is
The Katz score (edge-based) is
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10. RETURNING TO THE MATRIX
Carl Neumann
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11. Carl Neumann
I’ve heard the Neumann series called the “von Neumann”
series more than I’d like! In fact, the von Neumann kernel
of a graph should be named the “Neumann” kernel!
Wikipedia page
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12. PROPERTIES OF KATZ’S MATRIX
is symmetric
exists when
is spd when
Note that 1/max-degree suffices
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13. COMMUTE TIME
Picture taken from Google images, seems to be
Bay Bridge Traffic by Jim M. Goldstein.
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14. COMMUTE TIME
Consider a uniform random walk on a
graph
Also called the hitting
time from node i to j, or
the first transition time
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15. SKIPPING DETAILS
: graph Laplacian
is the only null-vector
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16. WHAT DO OTHER PEOPLE DO?
1) Just work with the linear algebra formulations
2) For Katz, Truncate the Neumann series as a few (3-5)
terms
3) Use low-rank approximations from EVD(A) or EVD(L)
4) For commute, use Johnson-Lindenstrauss inspired
random sampling
5) Approximately decompose into smaller problems
Liben-Nowell and Kleinberg CIKM2003, Acar et al. ICDM2009,
Spielman and Srivastava STOC2008, Sarkar and Moore UAI2007,Wang et al. ICDM2007
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17. THE PROBLEM
All of these techniques are
preprocessing based because
most people’s goal is to compute
all the scores.
We want to avoid
preprocessing the graph.
There are a few caveats here! i.e. one could solve the system instead of looking for the matrix inverse
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18. WHY NO PREPROCESSING?
The graph is constantly changing
as I rate new movies.
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19. WHY NO PREPROCESSING?
Top-k predicted “links”
are movies to watch!
Pairwise scores give
user similarity
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20. PAIRWISE ALGORITHMS
Katz
Commute
Golub and Meurant
to the rescue!
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21. MMQ - THE BIG IDEA
Quadratic form Think
Weighted sum A is s.p.d. use EVD
Stieltjes integral “A tautology”
Quadrature approximation
Matrix equation Lanczos
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22. LANCZOS
, k-steps of the Lanczos method
produce
and
=
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23. PRACTICAL LANCZOS
Only need to store the last 2 vectors in
Updating requires O(matvec) work
is not orthogonal
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24. MMQ PROCEDURE
Goal
Given
1. Run k-steps of Lanczos on starting with
2. Compute , with an additional eigenvalue at ,
set Correspond to a Gauss-Radau rule, with
u as a prescribed node
3. Compute , with an additional eigenvalue at , set
Correspond to a Gauss-Radau rule, with
l as a prescribed node
4. Output as lower and upper bounds on
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25. PRACTICAL MMQ
Increase k to become more accurate
Bad eigenvalue bounds yield worse results
and are easy to compute
not required, we can iteratively
update it’s LU factorization
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27. ONE LAST STEP FOR KATZ
Katz
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28. KATZ SCORES ARE LOCALIZED
Up to 50 neighbors is
99.65% of the total
mass
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29. TOP-K ALGORITHM FOR KATZ
Approximate
where is sparse
Keep sparse too
Ideally, don’t “touch” all of
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30. INSPIRATION - PAGERANK
Approximate
where is sparse
Keep sparse too? YES!
Ideally, don’t “touch” all of ? YES!
McSherry WWW2005, Berkhin 2007, Anderson et al. FOCS2008 – Thanks to Reid Anderson for telling me McSherry did this too.
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31. THE ALGORITHM - MCSHERRY
For
Start with the Richardson iteration
Rewrite
Richardson converges if
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32. THE ALGORITHM
Note is sparse.
If , then is sparse.
Idea
only add one component of to
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33. THE ALGORITHM
For
Init:
How to pick ?
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34. THE ALGORITHM FOR KATZ
For
Init:
Pick as max
Storing the non-zeros of the residual in a heap makes picking the max log(n) time. See Anderson et al. FOCS2008 for more
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35. CONVERGENCE?
The “standard” proof works for
1/max-degree.
For , then for symmetric ,
this algorithm is the Gauss-Southwell
procedure and it still converges.
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36. RESULTS – DATA, PARAMETERS
All unweighted, connected graphs
Easy
Hard
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41. CONCLUSIONS
These algorithms are faster than many
alternatives.
For pairwise commute, stopping criteria
are simpler with bounds
For top-k problems, we often need less
than 1 matvec for good enough results
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42. WARTS AND TODOS
Stopping criteria on our top-k algorithm
can be a bit hairy… we should refine it.
The top-k approach doesn’t work right for
commute time… we have an alternative.
requires the entire diagonal of
Take advantage of new research to “seed”
commute time better! von Luxburg et al. NIPS2010
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43. Paper at WAW2010
Slides should be online soon
Code is online already
stanford.edu/~dgleich/
publications/2010/codes/fast-katz
By AngryDogDesign on DeviantArt
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45. MAIN RESULTS – SLIDE ONE
A – adjacency matrix
L – Laplacian matrix
Katz score :
Commute time:
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46. MAIN RESULTS – SLIDE TWO
For Katz Compute one fast
Compute top fast
For Commute
Compute one fast
For almost commute
Compute top $F_{i,:}$ fast
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47. MAIN RESULTS – SLIDE THREE
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