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The Second Eigenvalue of Random Walks On Symmetric Random Intersection Graphs

  • Sotiris Nikoletseas
  • Christoforos Raptopoulos
  • Paul G. Spirakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4728)

Abstract

In this paper we examine spectral properties of random intersection graphs when the number of vertices is equal to the number of labels. We call this class symmetric random intersection graphs. We examine symmetric random intersection graphs when the probability that a vertex selects a label is close to the connectivity threshold τ c . In particular, we examine the size of the second eigenvalue of the transition matrix corresponding to the Markov Chain that describes a random walk on an instance of the symmetric random intersection graph G n, n, p. We show that with high probability the second eigenvalue is upper bounded by some constant ζ< 1.

Keywords

Random Walk Random Graph Hamilton Cycle Transition Probability Matrix Steady State Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. John Wiley & Sons, Inc., Chichester (2000)zbMATHGoogle Scholar
  2. 2.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Diaconis, P., Saloff-Coste, L.: Comparison Theorems for Reversible Markov Chains. The Annals of Applied Probability 3(3), 696–730 (1993)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Díaz, J., Penrose, M.D., Petit, J., Serna, M.: Approximating Layout Problems on Random Geometric Graphs. Journal of Algorithms 39, 78–116 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Díaz, J., Petit, J., Serna, M.: Random Geometric Problems on [0, 1]2. In: Rolim, J.D.P., Serna, M.J., Luby, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 294–306. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Díaz, J., Petit, J., Serna, M.: A Random Graph Model for Optical Networks of Sensors. In: The 1st International Workshop on Efficient and Experimental Algorithms (WEA) (2003), also in the IEEE Transactions on Mobile Computing Journal 2(3) 186–196 (2003)Google Scholar
  7. 7.
    Efthymiou, C., Spirakis, P.: On the Existence of Hamilton Cycles in Random Intersection Graphs. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 690–701. Springer, Heidelberg (2005)Google Scholar
  8. 8.
    Fill, J.A., Sheinerman, E.R., Singer-Cohen, K.B.: Random Intersection Graphs when m = ω(n): An Equivalence Theorem Relating the Evolution of the G(n, m, p) and G(n, p) models, http://citeseer.nj.nec.com/fill98random.html
  9. 9.
    Godehardt, E., Jaworski, J.: Two models of Random Intersection Graphs for Classification. In: Opitz, O., Schwaiger, M. (eds.) Studies in Classification, Data Analysis and Knowledge Organisation, pp. 67–82. Springer, Heidelberg (2002)Google Scholar
  10. 10.
    Jerrum, M., Sinclair, A.: Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains. Information and Computation 82, 93–133 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Karoński, M., Scheinerman, E.R., Singer-Cohen, K.B.: On Random Intersection Graphs: The Subgraph Problem. Combinatorics, Probability and Computing journal 8, 131–159 (1999)CrossRefGoogle Scholar
  12. 12.
    Marczewski, E.: Sur deux propriétés des classes d’ ensembles. Fund. Math. 33, 303–307 (1945)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Nikoletseas, S., Palem, K., Spirakis, P., Yung, M.: Short Vertex Disjoint Paths and Multiconnectivity in Random Graphs: Reliable Network Computing. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 508–519. Springer, Heidelberg (1994), also in the Special Issue on Randomized Computing of the International Journal of Foundations of Computer Science (IJFCS) 11(2), 247–262 (2000)Google Scholar
  14. 14.
    Nikoletseas, S., Raptopoulos, C., Spirakis, P.: The Existence and Efficient Construction of Large Independent Sets in General Random Intersection Graphs. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1029–1040. Springer, Heidelberg (2004), also in the Theoretical Computer Science (TCS) Journal (to appear, 2007)Google Scholar
  15. 15.
    Nikoletseas, S., Spirakis, P.: Expander Properties in Random Regular Graphs with Edge Faults. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 421–432. Springer, Heidelberg (1995)Google Scholar
  16. 16.
    Penrose, M.: Random Geometric Graphs. Oxford Studies in Probability (2003)Google Scholar
  17. 17.
    Raptopoulos, C., Spirakis, P.: Simple and Efficient Greedy Algorithms for Hamilton Cycles in Random Intersection Graphs. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 493–504. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Sinclair, A.: Algorithms for Random Generation and Counting: a Markov Chain Approach. PhD Thesis, University of Edimburg (1988)Google Scholar
  19. 19.
    Sinclair, A. (ed.): Algorithms for Random Generation and Counting. Birkhauser (1992)Google Scholar
  20. 20.
    Singer-Cohen, K.B.: Random Intersection Graphs. PhD thesis, John Hopkins University (1995)Google Scholar
  21. 21.
    Stark, D.: The Vertex Degree Distribution of Random Intersection Graphs. Random Structures & Algorithms 24(3), 249–258 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Sotiris Nikoletseas
    • 1
    • 2
  • Christoforos Raptopoulos
    • 1
    • 2
  • Paul G. Spirakis
    • 1
    • 2
  1. 1.Computer Technology Institute, P.O. Box 1122, 26110 PatrasGreece
  2. 2.University of Patras, 26500 PatrasGreece

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