What the matrix can tell us about the social network.
THE MATRIX & THESOCIAL NETWORKDAVID F. GLEICHPURDUE UNIVERSITY, COMPUTER SCIENCE
RÉSUMÉUndergraduate Harvey Mudd College Joint CS/Math!Internships Microsoft and Yahoo!!Graduate Stanford University ! "Computational and Mathematical Engineering!Internships Intel, Yahoo!, Microsoft, Library of Congress!Postdoc University of British Columbia!Postdoc Sandia National Laboratories "! "John von Neumann Fellow !!Faculty Purdue University Computer Science!!
The matrix is a powerful and productive paradigm for studying networks of connections.
Matrix computations 2 3 A1,1 A1,2 ··· A1,n 6 . 7 . 7 6 A2,1 A2,2 ··· . 7 A=6 . 6 7 4 . .. .. . . . Am 1,n 5 Am,1 ··· Am,n 1 Am,n Ax = b min kAx bk Ax = xLinear systems Least squares Eigenvalues
A new matrix-based sensitivity analysis of Google’s PageRank. Presented at" WAW2007, WWW2010 RAPr on Wikipedia E [x(A)] Std [x(A)] Published in the United States United States J. Internet Mathematics C:Living people C:Living people France C:Main topic classif. Led to new results on United Kingdom C:Contentsuncertainty quantiﬁcation in Germany C:Ctgs. by country physical simulations England United Kingdompublished in SIAM J. Matrix Canada France Analysis and SIAM J. Japan C:Fundamental Scientiﬁc Computing. Poland England Australia C:Ctgs. by topic Patent Pending Gleich (Stanford) Improved web-spam detection! Random sensitivity Ph.D. Defense 23 / 41 Collaborators Paul Constantine, Gianluca Iaccarino (physical simulation)
Fast matrix computations for " Tweet aloKatz scores and commute times. – MAIN RESULTS SLIDE THREE Presented at" WAW2010 Published in the J. Internet Mathematics Reduced computation time by orders of magnitude! David F. Gleich (Sandia) ICME la/opt seminarCollaborators Chen Greif, Laks V. S. Lakshmanan, Francesco Bonchi, Pooya Esfandiar
EK-HENG LIM · U. CHICAGO Reduction tree Tall-and-skinny QR factorizations (Red) S(2) The number of (Red) (Red) S(2) shuffleABILITY reducers and S(1) A on MapReduce architectures.matrix completion results are recovery the-ow when the solution of the convex heuristic solution. Using a recent matrix-completion iterations to use Network AlignmentDavid Gross (2010), we prove:Let s be centered, i.e., = 0. Let Y = sT e j David Gleich (Sandia) Iteration 1 MapReduce 2011 Iter 2 Iter 3 15/ere = m x s2 / (sT s) and = ((m x s ) Also, let r Algorithms for large sparse ⇢ H be a random set Square of elements O(2n (1 + )(log n)2 ) where = m x((n +en the solution of s kXk network alignment problems.o tr ce(X W ) = tr ce(( Y) W ), t W 2t with probability at least 1 n . A L BY “About n log n comparisons for recovery.” weight overlap theorem is not useful because we only need upper bound 60,120t of measurements from Y to generate the nstead, this theorem gives intuition for15,214 NetAlignBP 56,361 17,571 the solving an LP – 1 day iterative updates Overlapping clusters for distri- (think matrix-vectory probem. We test this by generating a skew- multiplies) – 10 min trix Y from a score vector s, and determine rounded LP 46,270mparisons we need before the we are able to 17,251 solving an LP –1 day buted network computations. ector s For these results, algorithm. We vertices, B is approx 300k vertices, and L is Note using the SVP A is approx 200k then do 5m edges. This setup yields a 5m variable integer QP.eriment by adding Gaussian noise to the mea-e ﬁnd aDavid F. Gleich (UBC) at n log n measurements. threshold Sandia 2 / 35 ry Noisy recovery 0.05 2n log(n) 6n log(n) Rank aggregation via skew- 0.04 Noise level 5n 0.03 0.02 symmetric matrix completion. 2n log(n) 6n log(n) 0.01 5n 3 4 0 10 10 200 1000 5000 Samples Samples C RESULTS
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