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A Nearly-Sublinear Method for Approximating a Column of the Matrix Exponential for Matrices from Large, Sparse Networks

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Algorithms and Models for the Web Graph (WAW 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8305))

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Abstract

We consider random-walk transition matrices from large social and information networks. For these matrices, we describe and evaluate a fast method to estimate one column of the matrix exponential. Our method runs in sublinear time on networks where the maximum degree grows doubly logarithmic with respect to the number of nodes. For collaboration networks with over 5 million edges, we find it runs in less than a second on a standard desktop machine.

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References

  1. Afanasjew, M., Eiermann, M., Ernst, O.G., Güttel, S.: Implementation of a restarted Krylov subspace method for the evaluation of matrix functions. Linear Algebra Appl. 429(10), 2293–2314 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersen, R., Chung, F., Lang, K.: Local graph partitioning using PageRank vectors. In: FOCS 2006 (2006)

    Google Scholar 

  4. Benzi, M., Boito, P.: Quadrature rule-based bounds for functions of adjacency matrices. Linear Algebra and its Applications 433(3), 637–652 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berkhin, P.: Bookmark-coloring algorithm for personalized PageRank computing. Internet Mathematics 3(1), 41–62 (2007)

    Article  MathSciNet  Google Scholar 

  6. Boguñá, M., Pastor-Satorras, R., Díaz-Guilera, A., Arenas, A.: Models of social networks based on social distance attachment. Phys. Rev. E 70(5), 056122 (2004)

    Article  Google Scholar 

  7. Bonchi, F., Esfandiar, P., Gleich, D.F., Greif, C., Lakshmanan, L.V.: Fast matrix computations for pairwise and columnwise commute times and Katz scores. Internet Mathematics 8(1-2), 73–112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chung, F.: The heat kernel as the PageRank of a graph. Proceedings of the National Academy of Sciences 104(50), 19735–19740 (2007)

    Article  Google Scholar 

  9. Estrada, E.: Characterization of 3d molecular structure. Chemical Physics Letters 319(5-6), 713–718 (2000)

    Article  Google Scholar 

  10. Estrada, E., Higham, D.J.: Network properties revealed through matrix functions. SIAM Review 52(4), 696–714 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Farahat, A., LoFaro, T., Miller, J.C., Rae, G., Ward, L.A.: Authority rankings from HITS, PageRank, and SALSA: Existence, uniqueness, and effect of initialization. SIAM Journal on Scientific Computing 27(4), 1181–1201 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13(5), 1236–1264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kondor, R.I., Lafferty, J.D.: Diffusion kernels on graphs and other discrete input spaces. In: ICML 2002, pp. 315–322 (2002)

    Google Scholar 

  15. Kunegis, J., Lommatzsch, A.: Learning spectral graph transformations for link prediction. In: Proceedings of the 26th Annual International Conference on Machine Learning, ICML 2009, pp. 561–568. ACM, New York (2009)

    Google Scholar 

  16. Luo, Z.Q., Tseng, P.: On the convergence of the coordinate descent method for convex differentiable minimization. J. Optim. Theory Appl. 72(1), 7–35 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45(1), 3–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Newman, M.E.J.: The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences 98(2), 404–409 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Newman, M.E.J.: Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74(3), 036104 (2006)

    Article  Google Scholar 

  20. Orecchia, L., Sachdeva, S., Vishnoi, N.K.: Approximating the exponential, the Lanczos method and an Õ(m)-time spectral algorithm for balanced separator. In: STOC 2012, pp. 1141–1160 (2012)

    Google Scholar 

  21. Sidje, R.B.: ExpoKit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24, 130–156 (1998)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, Purdue University, USA

    Kyle Kloster

  2. Computer Science Department, Purdue University, USA

    David F. Gleich

Authors
  1. Kyle Kloster
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  2. David F. Gleich
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Ryerson University, Toronto, ON, Canada

    Anthony Bonato  & Paweł Prałat  & 

  2. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    Michael Mitzenmacher

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Kloster, K., Gleich, D.F. (2013). A Nearly-Sublinear Method for Approximating a Column of the Matrix Exponential for Matrices from Large, Sparse Networks. In: Bonato, A., Mitzenmacher, M., Prałat, P. (eds) Algorithms and Models for the Web Graph. WAW 2013. Lecture Notes in Computer Science, vol 8305. Springer, Cham. https://doi.org/10.1007/978-3-319-03536-9_6

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  • DOI: https://doi.org/10.1007/978-3-319-03536-9_6

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03535-2

  • Online ISBN: 978-3-319-03536-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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