Commute times for a directed graph using an asymmetric Laplacian

Linear Algebra and Its Applications

Volumn 435, Issue 2, 2011, Pages 224-242

  Boley, Daniel ^a   Ranjan, Gyan ^a   Zhang, Zhi Li ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Commute times; Directed graphs; Graph Laplacians; Wireless networks

Indexed keywords

COMMUTE TIMES; DIRECTED GRAPHS; FUNDAMENTAL MATRIX; GRAPH LAPLACIANS; LAPLACIAN MATRICES; LAPLACIANS; RANDOM WALK; STRONGLY CONNECTED; UNDIRECTED GRAPH; WEIGHTED UNDIRECTED GRAPH;

GRAPH THEORY; MATRIX ALGEBRA; WIRELESS NETWORKS;

LAPLACE TRANSFORMS;

EID: 79952766932     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2011.01.030     Document Type: Article

References (40)

1. N. Alon Eigenvalues and expanders Combinatorica 6 2 1986 83 96
2. C. Berg, J. Christiansen, and P. Ressel Harmonic Analysis on Semigroups 1984 Springer Verlag
3. C.W. Borchardt Über eine der Interpolation entsprechende Darstellung der Eliminations-Resultante J. Reine Angew. Math. 57 1860 111 121
4. R.A. Brualdi, and H.J. Ryser Combinatorial Matrix Theory 1991 Cambridge University Press
5. A. Chandra, P. Raghavan, W. Ruzzo, R. Smolensky, P. Tiwari, The electrical resistance of a graph captures its commute and cover times, in: Proc. of Annual ACM Symposium on Theory of Computing, 1989, pp. 574-586.
6. C.K. Chau, P. Basu, Exact analysis of latency of stateless opportunistic forwarding, in: The 28th IEEE Conference on Computer Communications (IEEE INFOCOM 2009), 2010.
7. P. Chebotarev, E. Shamis, Matrix-forest theorems, February 2006, doi:arXiv:math/0602575v1.
8. F. Chung Spectral Graph Theory Am. Math. Soc. 1997
9. F. Chung Laplacians and the Cheeger inequality for directed graphs Ann. Comb. 9 1 2005 1 19 (Pubitemid 40550406)
10. W.E. Donath, and A.J. Hoffman Lower bounds for the partitioning of graphs IBM J. Res. Dev. 17 1973 410 425
11. P. Doyle, and J. Snell Random Walks and Electric Networks Math. Assoc. Am. 1984 front.math.ucdavis.edu/math.PR/0001057
12. M. Fiedler A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory Czechoslovak Math. J. 25 100 1975 619 633
13. J.A. Fill, and D.E. Fishkind The Moore-Penrose generalized inverse for sums of matrices SIAM J. Matrix Anal. Appl. 21 2 2000 629 635
14. F. Fouss, A. Pirotte, Renders J.M., and M. Saerens Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation IEEE Trans. Knowledge Data Eng. 19 2007 355 369 (Pubitemid 46374533)
15. F.R. Gantmacher Theory of Matrices vol. 2 1959 Chelsea New York
16. D. Gleich, Hierarchical directed spectral graph partitioning, Stanford Class Project, 2006. http://www.stanford.edu/∼dgleich/publications/directed- spectral.pdf
17. F. Gobel, and A. Jagers Random walks on graphs Stochastic Process. Appl. 2 1974 311 336
18. G.H. Golub, and C.F. Van Loan Matrix Computations third ed. 1996 Johns Hopkins University Press
19. C.M. Grinstead, and J.L. Snell Introduction to Probability second ed. 2006 American Mathematical Society www.dartmouth.edu/∼chance/teaching-aids/ books-articles/probability-book/book.html.
20. R.A. Horn, and C.R. Johnson Matrix Analysis 1985 Cambridge University Press Cambridge
21. S. Karlin, and H.M. Taylor A First Course in Stochastic Processes second ed. 1975 Academic Press New York
22. J. Kemeny, and J. Snell Finite Markov Chains 1976 Springer-Verlag
23. D. Klein, and M. Randic Resistance distance J. Math. Chem. 12 1993 81 95
24. Y. Li, Z.-L. Zhang, Random walks on digraphs: a theoretical framework for estimating transmission costs in wireless routing, in: The 29th IEEE Conference on Computer Communications (IEEE INFOCOM 2010), March 2010.
25. Y. Li, Z.-L. Zhang, Random walks on digraphs, the generalized digraph laplacian and the degree of asymmetry, in: 7th Workshop on Algorithms and Models for Webgraphs WAW'10, December 13-17 2010 (co-located with WINE'10). www.cs.umn.edu/∼zhzhang/Papers/Yanhua-Li-WAW10.pdf.
26. C. Meyer Generalized inversion of modified matrices SIAM J. Appl. Math. 24 1973 315 323
27. B. Mohar The Laplace spectrum of graphs Graph Theory, Combinatorics, and Applications vol. 2 1991 Wiley 871 898
28. B. Mohar Laplace eigenvalues of graphs - a survey Discrete Math. 109 1992 171 183
29. B. Mohar Some applications of Laplace eigenvalues of graphs Graph Symmetry: Algebraic Methods and Applications NATO ASI Ser C vol. 497 1997 Kluwer 225 275
30. A. Nilli On the second eigenvalue of a graph Discrete Math. 91 2 1991 207 210
31. J. Norris Markov Chains 1997 Cambridge University Press
32. G. Ranjan, Z.-L. Zhang, On random eccentricity in complex networks, Tech. Report, 2010. www-users.cs.umn.edu/∼granjan.
33. I.J. Schoenberg Remarks to Maurice Fréchet's article "Sur la définition axiomatique d'une classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert" Ann. Math. 36 3 1935 724 732
34. I.J. Schoenberg Metric spaces and positive definite functions Trans. Amer. Math. Soc. 44 1938 522 536
35. J. Shi, and J. Malik Normalised cuts and image segmentation IEEE Trans. Pattern Anal. Machine Intelligence 22 8 2000 888 905
36. R. Sinkhorn The range of the adjugate map Math. Mag. 66 2 1993 109 113
37. D.A. Spielman, and S.-H. Teng Spectral partitioning works:planar graphs and finite element meshes 37th Annual Symposium on Foundations of Computer Science 1996 IEEE Computer Soc. Press
38. D.W. Tutte The dissection of equilateral triangles into equilateral triangles Proc. Cambridge Phylos. Soc. 44 1948 463 482
39. U. von Luxburg A tutorial on spectral clustering Statist. Comput. 17 4 2007 395 416 Max Planck Institute for Biological Cybernetics, Technical Report No. TR-149
40. D. Zhou, J. Huang, B. Schölkopf, Learning from labeled and unlabeled data on a directed graph, in: Proc. 22nd Int'l Conf. Machine Learning, 2005, pp. 1041-1048. (Pubitemid 43183438)