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Computing Local and Global Centrality
- 1. COMPUTING LOCAL AND GLOBAL CENTRALITY DAVID F. GLEICH (AND MANY OTHERS)! DATA MINING, NETWORKS AND DYNAMICS 2011 NOVEMBER 7 1
- 2. LOCAL GLOBAL Pooya Esfandiar Reid Francesco Andersen Bonchi Chen Vahab Greif Mirrokni Laks V.S. Lakshmanan Byung- 2/41 Won On
- 3. Graph centrality Global How important is a node? Local How important is a node with respect to another one? 3/41
- 4. Graph centrality Koschützki et al. must respect isomorphism higher is better Examples node-degree 1/shortest-path 4/41
- 5. Graph centrality This talk Path summation X f (paths of length `) ` local Katz score X number of paths of ↵` · length ` between i and j ` 5/41
- 6. A – adjacency matrix L – Laplacian matrix P – random walk transition matrix Katz score Ki,j = [(I ↵AT ) 1 ]i,j Commute time Ci,j = vol(G)(L+ + L+ i,i j,j 2L+ ) i,j PageRank (I ↵P T )x = (1 ↵)e/n Xi,j = (1 ↵)[(I ↵P T ) 1 ]i,j 6/41
- 7. USES FOR CENTRALITY Ranking features for web-search/classification Najork, M. A.; Zaragoza, H. & Taylor, M. J.# HITS on the web: How does it compare? Becchetti, L.; Castillo, C.; Donato, D.; Baeza-Yates, R. & Leonardi, S. Link analysis for Web spam detection Interesting nodes GeneRank, ProteinRank, TwitterRank, IsoRank, FutureRank, HostRank, DiffusionRank, ItemRank, SocialPageRank, SimRank 7/41
- 8. USES FOR CENTRALITY Ranking networks of comparisons. Chartier, T. P.; Kreutzer, E.; Langville, A. N. & Pedings, K. E. Sensitivity and Stability of Ranking Vectors Clustering or community detection Andersen, R.; Chung, F. & Lang, K.# Local Graph Partitioning using PageRank Vectors Link prediction Savas et al. Hold on about 90 minutes 8/41
- 9. THESE GET USED A LOT. THEY MUST BE FAST. 9
- 10. MATRICES, MOMENTS, QUADRATURE Estimate a quadratic form T l x f (Z )x u T + (ei ej ) L (ei ej ) Commute 1 T 1 (ei + ej )T (I ↵P ) 1 (ei + ej ) (ei ej )T (I ↵P T ) 1 (ei ej ) Katz 4 4 Also used by Benzi and Bonito (LAA) for Katz scores and the matrix exponential 10/41
- 11. 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 David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 22 of 47 11/41
- 12. 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 12/41 David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 25 of 47
- 13. How well does it work? Bounds Error arxiv, Katz, hard alpha arxiv, Katz, hard 50 0 10 0 -5 10 -50 5 10 15 20 25 30 5 10 15 20 25 30 matrix-vector products matrix-vector products 13/41 ������ = 1/( || A ||2 + 1 )
- 14. MY COMPLAINTS Matvecs are expensive. Takes many iterations. Just one score comes out! 14/41
- 15. Katz scores ATZ SCORES ARE LOCALIZED T (I ↵A )k = e i are highly localized. Up to 50 neighbors is 99.65% of the total mass 15/41 Gleich (Purdue) Univ. Chicago SSCS Seminar 32 of 47
- 16. HOW CAN WE EXPLOIT THIS? 16
- 17. TOP-K ALGORITHM FOR KATZ Approximate T where is sparse Keep sparse too Ideally, don’t “touch” all of 17/41 David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 34 of
- 18. TOP-K ALGORITHM FOR KATZ Approximate T where is sparse Keep sparse too Ideally, don’t “touch” all of This is possible for " 18/41 David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 34 of personalized PageRank!
- 19. Richardson Ax = b x(k+1) = x(k) + r(k) A = AT , A ⌫ 0 Gradient descent r(k+1) = b Ax(k) equivalent# min xT Ax 2xT b to What about coordinate descent? Gauss-Southwell Ax = b x(k+1) = x(k) + rj(k) ej How to r(k+1) = r(k) + rj(k) Aej pick j? Frequently “rediscovered” for PageRank. 19/41 McSherry (WWW2005), Berkhin (JIM 2007), Andersen-Chung-Lang (FOCS 2006)
- 20. DEMO! 20
- 21. NEW CONVERGENCE THEORY Katz and PageRank are equivalent if ������ < 1 / || A ||1 Gauss-Southwell converges when ������ < 1 / || A ||2 (Luo and Tseng 1992) if j is picked as the largest residual Read all about it Fast matrix computations for pair-wise and column-wise commute times and Katz scores. Bonchi, Esfandiar, Gleich, Greif, Lakshmanan, J. Internet Mathematics (to appear) 21/41
- 22. 1,000,000 node, 100,000,000 edges hollywood, Katz, hard alpha Precision@k for exact top−k sets 1 0.8 0.6 0.4 k=10 k=100 0.2 k=1000 cg k=25 0 k=25 −2 −1 0 1 2 22/41 10 10 10 10 10 Equivalent matrix−vector products
- 23. OPEN QUESTIONS I can’t find any existing derivation of this method in the non-symmetric case (prior to the PageRank literature). Any thoughts? How to show that the method convergence for a non-symmetric matrix when (I ↵P T ) is not diagonally dominant? 23/41
- 25. LARGE GRAPHS, IN PRACTICE Copy 1 Copy 2 src -> dst src -> dst src -> dst src -> dst src -> dst src -> dst Copy 1 Copy 2 src -> dst src -> dst src -> dst src -> dst src -> dst src -> dst Copy 1 Copy 2 src -> dst src -> dst src -> dst src -> dst src -> dst src -> dst Edge lists maybe tied together by a 25/41 common host, stored redundantly on many hard drives.
- 26. UTILIZE SOME REDUNDANCY? To compute global PageRank? 26
- 27. Overlapping Clusters Use the redundancy to reduce communication when solving a PageRank problem Overlapping clusters for distributed computation. # 27/41 Andersen, Gleich, Mirrokni, WSDM2012 (to appear).
- 28. Communication avoiding algorithms Communication is the limiting factor in most computations these days. Flops are, relatively speaking, free. 28/41
- 29. KEY POINTS Utilize personalized PageRank vectors to find the clusters with “good” conductance scores. Define “core” vertices for each cluster. Find a good way to cover the graph with these clusters. Use restricted additive Schwarz to solve # (thanks Prof. Szyld and Frommer!) 29/41
- 30. All nodes solve locally using # the coordinate descent method. 30/41
- 31. All nodes solve locally using # the coordinate descent method. A core vertex for the 31/41 gray cluster.
- 32. All nodes solve locally using # the coordinate descent method. Red sends residuals to white. White send residuals to red. 32/41
- 33. White then uses the coordinate descent method to adjust its solution. 33/41 Will cause communication to red/blue.
- 34. It works! 2 Swapping Probability (usroads) PageRank Communication (usroads) Swapping Probability (web−Google) 1.5 PageRank Communication (web−Google) Relative Work 1 Metis Partitioner 0.5 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Volume Ratio How much more of the 34/41 graph we need to store.
- 35. PERSONALIZED PAGERANK CLUSTERS Solve (I ↵P T )x = (1 ↵)ei # to a large degree-weighted tolerance ������ Sweep over the vertices in order of their degree- normalized rank. Find the best conductance set. A Cheeger-like inequality. (Not a heuristic.) 35/41
- 36. CORE VERTICES Compute the expected “leavetime” for each vertex in a cluster. Keep increasing the threshold for a “good” vertex until every vertex is core in some cluster. Then approximate a set-cover problem to cover the graph with clusters, and use a heuristic to pack vertices until 36/41
- 37. MY QUESTIONS " and future directions REVERSE ORDER 37
- 38. GRAPH SPECTRA 38/41 Some work by Banerjee and Jost.