Locally supported eigenvectors of matrices associated with connected and unweighted power-law graphs

Electronic Transactions on Numerical Analysis

Volumn 39, Issue , 2012, Pages 353-378

  Henson, Van Emden ^a   Sanders, Geoffrey ^a  

^a LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

Author keywords

Adjacency matrix; Eigenvalues; Eigenvectors; Graph Laplacian; Sparse matrices

Indexed keywords

GRAPH STRUCTURES;

ADJACENCY MATRICES; COMPUTATIONAL SAVINGS; EIGENVALUES; GRAPH LAPLACIAN; GRAPH LAPLACIANS; POWER LAW GRAPHS; PRE-PROCESSING STEP; SPARSE MATRICES;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84870820592     PISSN: None     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (28)

1. P. ARBENZ, U. HETMANIUK, R. LEHOUCQ, AND R. TUMINARO, A comparison of eigensolvers for largescale 3D modal analysis using AMG-preconditioned iterative methods, Internat. J. Numer. Methods Engrg., 64 (2005), pp. 204-236.
2. A. BANERJEE AND J. JOST, On the spectrum of the normalized graph Laplacian, Linear Algebra Appl., 428 (2008), pp. 3015-3022.
3. A.-L. BARABÁSI AND R. ALBERT, Emergence of scaling in random networks, Science, 286 (1999), pp. 509-512.
4. V. BATAGELJ AND M. ZAVERSNIK, An O(m) algorithm for cores decomposition of networks, preprint, CoRR, cs.DS/0310049, 2003. http://arxiv.org/abs/cs.DS/ 0310049.
5. T. BIYIKOGU, J. LEYDOLD, AND P. STADLER, Laplacian Eigenvectors of Graphs, Lecture Notes in Mathematics 1915, Springer, Heidelberg, 2007.
6. S. BRIN AND L. PAGE, The anatomy of a large-scale hypertextual web search engine, Comput. Networks ISDN Systems, 30 (1998), pp. 107-117.
7. F. BONCHI, P. ESFANDIAR, D. GLEICH, C. GRIEF, AND L. LAKSHAMANAN, Fast matrix computations for pairwise and columnwise commute times and Katz scores, Internet Math., 8 (2012), pp. 73-112.
8. A. BORZÍ AND G. BORZÍ, Algebraic multigrid methods for solving generalized eigenvalue problems, Internat. J. Numer. Methods Engrg., 65 (2006), pp. 1186-1196.
9. I. FARIA, Permanental roots and the star degree of a graph, Linear Algebra Appl., 64 (1985), pp. 255-265.
10. F. FOUSS, A. PIROTTE, J. RENDERS, AND M. SAERENS, Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation, IEEE Trans. Knowl. Data Engrg., 19 (2007), pp. 355-369.
11. D. Gleich, The Spectre of the Spectrum, Lecture given at the Indiana University, Bloomington, IN, and personal correspondence, 2011.
12. V. HERNANDEZ, J. ROMAN, AND V. VIDAL, SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31 (2005), pp. 351-362.
13. M. HEROUX, R. BARTLETT, V. HOWLE, R. HOEKSTRA, J. HU, T. KOLDA, R. LEHOUCQ, K. LONG, R. PAWLOWSKI, E. PHIPPS, A. SALINGER, H. THORNQUIST, R. TUMINARO, J. WILLENBRING, AND A. WILLIAMS, An Overview of Trilinos, Sand Report, SAND2003-2927, Sandia National Laboratories, Albuquerque, 2003.
14. U. HETMANIUK, A Rayleigh quotient minimization algorithm based on algebraic multigrid, Numer. Linear Algebra Appl., 14 (2007), pp. 563-580.
15. Y. KOREN, Drawing graphs by eigenvectors: theory and practice, Comput.Math. Appl., 49 (2005), pp. 1867-1888.
16. D. KUSHNIR, M. GALUN, AND A. BRANDT, Efficient multilevel eigensolvers with applications to data analysis tasks, IEEE Trans. Pattern Anal. Mach. Intell., 32 (2010), pp. 1377-1391.
17. I. LASHUK, M. E. ARGENTATI, E. OVCHINNIKOV, AND A. V. KNYAZEV, Preconditioned eigensolver LOBPCG in hypre and PETSc, in Proceedings of the 16th International Conference on Domain Decomposition Methods 2005, O. B. Widlund and D. E. Keyes, eds., Lect. Notes Comput. Sci. Eng., 55, Springer, Berlin, 2007, pp. 635-642.
18. J. LESKOVEC, Stanford network analysis platform. http://snap.stanford. edu.
19. B. LYON, The Opte project. http://www.opte.org.
20. R. MERRIS, Laplacian matrices of graphs: a survey, Linear Algebra Appl., 197/198 (1994), pp. 143-176.
21. -, Laplacian graph eigenvectors, Linear Algebra Appl., 278 (1998), pp. 221-236.
22. M. NEWMAN, The Laplacian spectrum of graphs, Ph.D. Thesis, Dept. of Math., University of Manitoba, Winnipeg, Canada, 2000.
23. M. E. J. NEWMAN, Networks: An Introduction, Oxford University Press, Oxford, 2010.
24. B. PARLETT, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, 1998.
25. C. TSOURAKAKIS, Fast counting of triangles in large real networks: algorithms and laws, in ICDM '08: Proceedings of the 2008 Eighth IEEE International Conference on Data Mining 2008, G. Fosca, ed., IEEE Conference Publication, IEEE, Piscataway, NJ, 2008, pp. 608-617.
26. P. VANEK, J. MANDEL, AND M. BREZINA, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179-196.
27. U. VON LUXBURG, A tutorial on spectral clustering, Stat. Comput., 17 (2007), pp. 395-416.
28. U. VON LUXBURG, A. RADL, AND M. HEIN, Getting lost in space: large sample analysis of the commute distance, in Advances in Neural Information Processing Systems 23, J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel and A. Culotta, eds., NIPS Foundation, La Jolla, CA, 2010, pp. 2622-2630.