Expansion and lack thereof in randomly perturbed graphs

Internet Mathematics

Volumn 4, Issue 2-3, 2007, Pages 131-147

  Flaxman, Abraham D ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79956048530     PISSN: 15427951     EISSN: None     Source Type: Journal    
DOI: 10.1080/15427951.2007.10129290     Document Type: Article

References (43)

1. N. Alon. “Eigenvalues and Expanders.” Combinatorica 6:2 (1986), 83–96.
2. N. Alon and N. Kahale. “A Spectral Technique for Coloring Random 3-Colorable Graphs.” SIAM J. Comput. 26:6 (1997), 1733–1748.
3. N. Alon and V. D. Milman. “λ1, Isoperimetric Inequalities for Graphs, and Superconcentrators.” J. Combin. Theory Ser. B 38:1 (1985), 73–88.
4. N. Alon and J. H. Spencer. The Probabilistic Method, Second edition, Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: Wiley-Interscience, 2000.
5. R. Andersen, F. Chung, and L. Lu. “Analyzing the Small World Phenomenon Using a Hybrid Model with Local Network Flow.” In Algorithms and Models for the Web-Graph: Third International Workshop, WAW 2004, Rome, Italy, October 16, 2004, Proceeedings, Lecture Notes in Computer Science 3243, pp. 19–30. Berlin: Springer, 2004.
6. C. Banderier, R. Beier, and K. Mehlhorn. “Smoothed Analysis of Three Combinatorial Problems.” In Mathematical Foundations of Computer Science 2003: 28th International Symposium, MFCS 2003, Bratislava, Slovakia, August 25-29, 2003, Proceedings, Lecture Notes in Computer Science 2747, pp. 198–207. Berlin: Springer, 2003.
7. L. Becchetti, S. Leonardi, A. Marchetti-Spaccamela, G. Schäfer, and T. Vredeveld. “Average Case and Smoothed Competitive Analysis of the Multi-Level Feedback Algorithm.” In Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS 2003), p. 462. Washington, DC: IEEE Computer Society, 2003.
8. R. Beier and B. Vöcking. “Typical Properties of Winners and Losers in Discrete Optimization.” In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, June 13–15, 2004, pp. 343–352. New York: ACM Press, 2004.
9. I. Benjamini and N. Berger. “The Diameter of Long-Range Percolation Clusters on Finite Cycles.” Random Structures Algorithms 19:2 (2001), 102–111.
10. M. Biskup. “On the Scaling of the Chemical Distance in Long-Range Percolation Models.” Ann. Probab. 32:4 (2004), 2938–2977.
11. D. K. Blandford, G. E. Blelloch, and I. A. Kash. “Compact Representations of Separable Graphs.” In Proceedings of the Fourteenth Annual ACM SIAM Symposium on Discrete Algorithms, Baltimore, Maryland, January 12–14, 2003, pp. 679–688. Philadelphia, PA: SIAM, 2003.
12. T. Bohman, A. Frieze, and R. Martin. “How Many Random Edges Make a Dense Graph Hamiltonian?” Random Structures Algorithms 22:1 (2003), 33–42.
13. T. Bohman, A. Frieze, M. Krivelevich, and R. Martin. “Adding Random Edges to Dense Graphs.” Random Structures Algorithms 24:2 (2004), 105–117.
14. B. Bollobás and F. R. K. Chung. “The Diameter of a Cycle Plus a Random Matching.” SIAM J. Discrete Math. 1:3 (1988), 328–333.
15. J. Cheeger. “A Lower Bound for the Smallest Eigenvalue of the Laplacian.” In Problems in Analysis: A Symposium in Honor of Salomon Bochner, edited by R. C. Gunning, pp. 195–199. Princeton, NJ: Princeton University Press, 1970.
16. F. Chung. “Laplacians and the Cheeger Inequality for Directed Graphs.” Ann. Comb. 9:1 (2005), 1–19.
17. F. Chung and L. Lu. “The Small World Phenomenon in Hybrid Power Law Graphs.” In Complex Networks, Lecture Notes in Physics 650, pp. 89–104. Berlin: Springer, 2004.
18. F. Chung, L. Lu, and V. Vu. “Eigenvalues of Random Power Law Graphs.” Ann. Comb. 7:1 (2003), 21–33.
19. C. Cooper and A. Frieze. “Crawling on Simple Models of Web Graphs.” Internet Math. 1:1 (2003), 57–90.
20. D. Coppersmith, D. Gamarnik, and M. Sviridenko. “The Diameter of a Long-Range Percolation Graph.” Random Structures Algorithms 21:1 (2002), 1–13.
21. A. Deshpande and D. A. Spielman. “Improved Smoothed Analysis of the Shadow Vertex Simplex Method.” In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 349–356. Washington, DC: IEEE Computer Society, 2005.
22. E. Estrada. “Spectral Scaling and Good Expansion Properties in Complex Networks.” Europhysics Letters 73:4 (2006), 649–655.
23. M. Faloutsos, P. Faloutsos, and C. Faloutsos. “On Power-Law Relationships of the Internet Topology.” In SIGCOMM’99: Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, pp. 251–262. New York: ACM Press, 1999.
24. U. Feige and E. Ofek. “Spectral Techniques Applied to Sparse Random Graphs.” Random Structures and Algorithms 27:2 (2005), 251–275.
25. A. D. Flaxman and A. M. Frieze. “The Diameter of Randomly Perturbed Digraphs and Some Applications.” In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques: Proceedings of the 8th International Workshop on Randomization and Computation, RANDOM 2004, Cambridge, MA, USA, August 2004, Lecture Notes in Computer Science 3122, pp. 345–356, edited by K. Jansen, S. Khanna, J. D. P. Rolim, and D. Ron, Berlin: Springer, 2004.
26. A. Flaxman, A. M. Frieze, and J. Vera. “A Geometric Preferential Attachment Model of Networks.” In Algorithms and Models for the Web-Graph: Third International Workshop, WAW 2004, Rome, Italy, October 16, 2004, Proceeedings, Lecture Notes in Computer Science 3243, pp. 44–55. Berlin: Springer, 2004.
27. A. Flaxman, A. Frieze, and T. Fenner. “High Degree Vertices and Eigenvalues in the Preferential Attachment Graph.” Internet Math. 2:1 (2005), 1–19.
28. J. Friedman. “A Proof of Alon’s Second Eigenvalue Conjecture.” In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pp. 720–724. New York: ACM Press, 2003.
29. J. Friedman, J. Kahn, and E. Szemerédi. “On the Second Eigenvalue of Random Regular Graphs.” In Proceedings of the Twenthy-First Annual ACM Symposium on Theory of Computing, Seattle, Washington, May 15–17, 1989, pp. 587–598. New York: ACM Press, 1989.
30. Z. Füredi and J. Komlós. “The Eigenvalues of Random Symmetric Matrices.” Combinatorica 1:3 (1981), 233–241.
31. S. Janson, T. _Luczak, and A. Rucinski. Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: Wiley-Interscience, 2000.
32. F. Juhász. “On the Spectrum of a Random Graph.” In Algebraic Methods in Graph Theory, Colloquia Mathematica Societatis Janos Bolyai 25, pp. 313–316. Amsterdam: North-Holland, 1981.
33. J. Kleinberg. “The Small-World Phenomenon: An Algorithm Perspective.” In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21–23, 2000, Portland, OR, USA, pp. 163–170. New York: ACM Press, 2000.
34. M. Krivelevich, B. Sudakov, and P. Tetali. “On Smoothed Analysis in Dense Graphs and Formulas.” Random Structures Algorithms 29:2 (2006), 180–193.
35. G. F. Lawler and A. D. Sokal. “Bounds on the L2 Spectrum for Markov Chains and Markov Processes: A Generalization of Cheeger’s Inequality.” Trans. Amer. Math. Soc. 309:2 (1988), 557–580.
36. M. Mihail and C. Papadimitriou. “On the Eigenvalue Power Law.” In Randomization and Approximation Techniques in Computer Science, Lecture Notes in Computer Science 2483, pp. 254–262, Berlin: Springer, 2002.
37. M. Mihail, C. Papadimitriou, and A. Saberi. “On Certain Connectivity Properties of the Internet Topology.” In Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS 2003), p. 28. Washington, DC: IEEE Computer Society, 2003.
38. R. Monasson. “Diffusion, Localization and Dispersion Relations on ‘Small-World’ Lattices.” The European Physical Journal B 12, 555–567 (1999).
39. A. Sinclair and M. Jerrum. “Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains.” Inform. and Comput. 82:1 (1989), 93–133.
40. D. Spielman and S.-H. Teng. “Smoothed Analysis of Algorithms: Why the Simplex Algorithm Usually Takes Polynomial Time.” In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, July 6–8, 2001, Heraklion, Crete, Greece, pp. 296–305, New York: ACM Press, 2001.
41. D. A. Spielman and S.-H. Teng. “Smoothed Analysis: Motivation and Discrete Models.” In Algorithms and Data Structures, Lecture Notes in Computer Science 2748, pp. 256–270, Berlin: Springer, 2003.
42. D. J. Watts and S. H. Strogatz. “Collective Dynamics of “Small-World Networks.” Nature 292 (1998), 440–442.
43. C. W. Wu. “Algebraic Connectivity of Directed Graphs.” Linear Multilinear Algebra 53:3 (2005), 203–223.