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Fast Random Walk with Restart and Its Applications

Fast Random Walk with Restart and Its Applications. Hanghang Tong, Christos Faloutsos and Jia-Yu (Tim) Pan. ICDM 2006 Dec. 18-22, HongKong. Motivating Questions. Q: How to measure the relevance?

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Fast Random Walk with Restart and Its Applications

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  1. Fast Random Walk with Restart and Its Applications Hanghang Tong, Christos Faloutsos and Jia-Yu (Tim) Pan ICDM 2006 Dec. 18-22, HongKong

  2. Motivating Questions • Q: How to measure the relevance? • A: Random walk with restart • Q: How to do it efficiently? • A: This talk tries to answer!

  3. 10 9 12 2 8 1 11 3 4 6 5 7 Random walk with restart

  4. 0.03 0.04 10 9 0.10 12 2 0.08 0.02 0.13 8 1 0.13 11 3 0.04 4 0.05 6 5 0.13 7 0.05 Random walk with restart Nearby nodes, higher scores Ranking vector More red, more relevant

  5. { } Cat Forest Grass Tiger {?, ?, ?,} Automatic Image Caption • Q … { } Sea Sun Sky Wave ? A: RWR! [Pan KDD2004]

  6. Sea Sun Sky Wave Cat Forest Tiger Grass Region Image Test Image Keyword

  7. Region Image Test Image {Grass, Forest, Cat, Tiger} Sea Sun Sky Wave Cat Forest Tiger Grass Keyword

  8. Neighborhood Formulation … … Q: what is most related conference to ICDM A: RWR! [Sun ICDM2005] … … Conference Author

  9. NF: example

  10. Center-Piece Subgraph(CePS) Q ? Original Graph Black: query nodes CePS A: RWR! [Tong KDD 2006]

  11. CePS: Example

  12. Other Applications • Content-based Image Retrieval [He] • Personalized PageRank [Jeh], [Widom], [Haveliwala] • Anomaly Detection (for node; link) [Sun] • Link Prediction [Getoor], [Jensen] • Semi-supervised Learning [Zhu], [Zhou] • …

  13. Roadmap • Background • RWR: Definitions • RWR: Algorithms • Basic Idea • FastRWR • Pre-Compute Stage • On-Line Stage • Experimental Results • Conclusion

  14. 10 9 12 2 8 1 11 3 4 6 5 7 Computing RWR Restart p Starting vector Adjacent matrix Ranking vector 1 n x 1 n x n n x 1

  15. : Maxwell Equation for Web! Beyond RWR [Chakrabarti] P-PageRank [Haveliwala] SM Learning [Zhou, Zhu] RL in CBIR [He] RWR [Pan, Sun] PageRank [Haveliwala] Fast RWR Finds the Root Solution !

  16. Q: Given query i, how to solve it? ? ?

  17. 10 9 12 2 8 1 11 3 0.04 0.03 10 9 0.10 12 4 0.13 0.08 2 0.02 8 1 11 0.13 3 6 0.04 5 4 0.05 6 5 0.13 7 7 0.05 OntheFly: No pre-computation/ light storage Slow on-line response O(mE)

  18. 0.04 0.03 10 9 0.10 12 0.13 0.08 2 0.02 8 1 11 0.13 3 0.04 4 0.05 6 5 0.13 7 0.05 PreCompute 10 9 12 2 8 1 11 R: 3 4 6 5 7 [Haveliwala]

  19. 10 9 12 2 8 1 11 3 0.04 0.03 10 9 0.10 12 4 0.13 0.08 2 0.02 8 1 11 0.13 3 6 0.04 5 4 0.05 6 5 0.13 7 7 0.05 PreCompute: Fast on-line response Heavy pre-computation/storage cost O(n ) 3 O(n ) 2

  20. Q: How to Balance? On-line Off-line

  21. Roadmap • Background • RWR: Definitions • RWR: Algorithms • Basic Idea • FastRWR • Pre-Compute Stage • On-Line Stage • Experimental Results • Conclusion

  22. 10 10 9 9 12 12 2 2 8 8 1 1 11 11 3 3 4 4 10 10 9 9 6 6 5 5 12 12 2 2 8 8 1 1 11 11 7 7 3 3 4 4 6 6 5 5 7 7 0.04 10 10 0.03 9 9 10 9 12 12 0.10 2 2 12 0.13 0.08 2 8 8 1 1 0.02 11 11 8 3 3 1 11 0.13 3 0.04 4 4 4 6 6 5 5 0.05 6 5 0.13 7 7 7 0.05 Basic Idea Find Community Combine Fix the remaining

  23. Pre-computational stage -1 • Q: • A: A few small, instead of ONE BIG, matrices inversions Efficiently compute and store Q

  24. On-Line Query Stage -1 • Q: Efficiently recover one column of Q • A: A few, instead of MANY, matrix-vector multiplication +

  25. Roadmap • Background • RWR: Definitions • RWR: Algorithms • Basic Idea • FastRWR • Pre-Compute Stage • On-Line Stage • Experimental Results • Conclusion

  26. Pre-compute Stage • p1: B_Lin Decomposition • P1.1 partition • P1.2 low-rank approximation • p2: Q matrices • P2.1 computing (for each partition) • P2.2 computing (for concept space)

  27. 10 9 12 2 8 1 11 3 4 6 5 7 P1.1: partition 10 9 12 2 8 1 11 3 4 6 5 7 Within-partition links cross-partition links

  28. 10 9 12 2 8 1 11 3 4 6 5 7 P1.1: block-diagonal 10 9 12 2 8 1 11 3 4 6 5 7

  29. 10 9 12 2 8 1 11 3 4 6 5 7 P1.2: LRA for 10 9 12 2 8 1 11 3 4 6 5 7 ~ |S| << |W2|

  30. = +

  31. p2.1 Computing

  32. Q 1,1 1,2 Q Q 1,k Comparing and • Computing Time • 100,000 nodes; 100 partitions • Computing 100,00x is Faster! • Storage Cost • 100x saving! =

  33. ~ • Q: How to fix the green portions? ~ + ~ ? +

  34. Q 1,1 1,2 Q 10 9 Q 1,k 12 2 8 1 11 3 4 6 5 7 p2.2 Computing: -1 _ U = V

  35. We have: Communities Bridges SM Lemma says:

  36. Roadmap • Background • RWR: Definitions • RWR: Algorithms • Basic Idea • FastRWR • Pre-Compute Stage • On-Line Stage • Experimental Results • Conclusion

  37. On-Line Stage • Q ? + Query Result Pre-Computation • A (SM lemma)

  38. q1: q2: q3: q4: q5: q6: On-Line Query Stage

  39. Roadmap • Background • RWR: Definitions • RWR: Algorithms • Basic Idea • FastRWR • Pre-Compute Stage • On-Line Stage • Experimental Results • Conclusion

  40. Experimental Setup • Dataset • DBLP/authorship • Author-Paper • 315k nodes • 1,800k edges • Approx. Quality: Relative Accuracy • Application: Center-Piece Subgraph

  41. Query Time vs. Pre-Compute Time Log Query Time • Quality: 90%+ • On-line: • Up to 150x speedup • Pre-computation: • Two orders saving Log Pre-compute Time

  42. Query Time vs. Pre-Storage Log Query Time • Quality: 90%+ • On-line: • Up to 150x speedup • Pre-storage: • Three orders saving Log Storage

  43. Roadmap • Background • RWR: Definitions • RWR: Algorithms • Basic Idea • FastRWR • Pre-Compute Stage • On-Line Stage • Experimental Results • Conclusion

  44. Conclusion • FastRWR • Reasonable quality preservation (90%+) • 150x speed-up: query time • Orders of magnitude saving: pre-compute & storage • More in the paper • The variant of FastRWR and theoretic justification • Implementation details • normalization, low-rank approximation, sparse • More experiments • Other datasets, other applications

  45. Q&A Thank you! htong@cs.cmu.edu www.cs.cmu.edu/~htong

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