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Fast Direction-Aware Proximity for Efficient Graph Mining

Learn about defining Direction-Aware Proximity (DAP) for graph mining efficiency, overcoming matrix inversion challenges, computing proximities on medium and large graphs, FastAllDAP and FastOneDAP algorithms, link prediction using proximity, with practical modifications for node degree and weakly connected pairs.

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Fast Direction-Aware Proximity for Efficient Graph Mining

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  1. FastDirection-Aware Proximity for Graph Mining KDD 2007, San Jose Hanghang Tong, Yehuda Koren, Christos Faloutsos

  2. Defining Direction-Aware Proximity (DAP): escape probability • Define Random Walk (RW) on the graph • Esc_Prob(AB) • Prob (starting at A, reaches B before returning toA) the remaining graph A B Esc_Prob = Pr (smile before cry)

  3. I - P= P: Transition matrix (row norm.) -1 Esc_Prob(1->5) = +

  4. Intuition of Formula P*P=

  5. I - P= P: Transition matrix (row norm.) -1 Esc_Prob(1->5) = +

  6. Challenges • Case 1, Medium Size Graph • Matrix inversion is feasible, but… • What if we want many proximities? • Q: How to get all (n ) proximities efficiently? • A: FastAllDAP! • Case 2: Large Size Graph • Matrix inversion is infeasible • Q: How to get one proximity efficiently? • A: FastOneDAP! 2

  7. FastAllDAP • Q1: How to efficiently compute all possible proximities on a medium size graph? • a.k.a. how to efficiently solve multiple linear systems simultaneously? • Goal: reduce # of matrix inversions!

  8. FastAllDAP: Observation P= P= Need two different matrix inversions!

  9. FastAllDAP: Rescue Prox(1  5) P= Prox(1  6) Overlap between two gray parts! P= Redundancy among different linear systems!

  10. FastAllDAP: Theorem • Example: • Theorem: • Proof: by SM Lemma

  11. FastAllDAP: Algorithm • Alg. • Compute Q • For i,j =1,…, n, compute • Computational Save O(1) instead of O(n )! • Example • w/ 1000 nodes, • 1m matrix inversion vs. 1 matrix! 2

  12. FastOneDAP • Q1: How to efficiently compute one single proximity on a large size graph? • a.k.a. how to solve one linear system efficiently? • Goal: avoid matrix inversion!

  13. FastOneDAP: Observation Partial Info. (4 elements /2 cols ) of Q is enough!

  14. Reminder: T [0, …0, 1, 0, …, 0] th i col of Q FastOneDAP: Observation • Q: How to compute one column of Q? • A: Taylor expansion

  15. T [0, …0, 1, 0, …, 0] th i col of Q FastOneDAP: Observation …. x x x Sparse matrix-vector multiplications!

  16. Alg. to estimate i Col of Q FastOneDAP: Iterative Alg. th

  17. Convergence Guaranteed ! Computational Save Example: 100K nodes and 1M edges (50 Iterations) 10,000,000x fast! Footnote: 1 col is enough! (details in paper) FastOneDAP: Property

  18. Esc_Prob is good, but… • Issue #1: • `Degree-1 node’ effect • Issue #2: • Weakly connected pair Need some practical modifications!

  19. Issue#1: `degree-1 node’ effect[Faloutsos+] [Koren+] • no influence for degree-1 nodes (E, F)! • known as ‘pizza delivery guy’ problem in undirected graph • Solutions: Universal Absorbing Boundary! Esc_Prob(a->b)=1 Esc_Prob(a->b)=1

  20. Universal Absorbing Boundary U-A-B is a black-hole! Footnote: fly-out probability = 0.1

  21. Introducing Universal-Absorbing-Boundary Esc_Prob(a->b)=1 Prox(a->b)=0.91 Esc_Prob(a->b)=1 Prox(a->b)=0.74 Footnote: fly-out probability = 0.1

  22. Issue#2: Weakly connected pair Prox(AB) = Prox (BA)=0 Solution: Partial symmetry!

  23. Practical Modifications: Partial Symmetry Prox(AB) = Prox (BA)=0 Prox(AB) =0.081 > Prox (BA)=0.009

  24. Efficiency: FastAllDAP Time (sec) Straight-Solver 1,000x faster! FastAllDAP Size of Graph

  25. Efficiency: FastOneDAP Time (sec) Straight-Solver 1,0000x faster! FastOneDAP Size of Graph

  26. Link Prediction: direction • Q: Given the existence of the link, what is the direction of the link? • A: Compare prox(ij) and prox(ji) >70% density Prox (ij) - Prox (ji)

  27. Thanks. Any Question?

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