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A Sequential Algorithm for Generating Random Graphs

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2007, RANDOM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

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Abstract

We present the fastest FPRAS for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence \((d_i)_{i=1}^n\) with maximum degree \(d_{\max}=O(m^{1/4-\tau})\), our algorithm generates almost uniform random graph with that degree sequence in time O(m d max ) where is the number of edges in the graph and τ is any positive constant. The fastest known FPRAS for this problem [22] has running time of O(m 3 n 2). Our method also gives an independent proof of McKay’s estimate [33] for the number of such graphs.

Our approach is based on sequential importance sampling (SIS) technique that has been recently successful for counting graphs [15,11,10]. Unfortunately validity of the SIS method is only known through simulations and our work together with [10] are the first results that analyze the performance of this method.

Moreover, we show that for d = O(n 1/2 − τ), our algorithm can generate an asymptotically uniform d-regular graph. Our results are improving the previous bound of d = O(n 1/3 − τ) due to Kim and Vu [30] for regular graphs.

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References

  1. Alon, N., Spencer, J.: The Probabilistic Method. Wiley, NewYork (1992)

    MATH  Google Scholar 

  2. Alderson, D., Doyle, J., Willinger, W.: Toward and Optimization-Driven Framework for Designing and Generating Realistic Internet Topologies. HotNets (2002)

    Google Scholar 

  3. Amraoui, A., Montanari, A., Urbanke, R.: How to Find Good Finite-Length Codes: From Art Towards Science (preprint, 2006), arxiv.org/pdf/cs.IT/0607064

  4. Bassetti, F., Diaconis, P.: Examples Comparing Importance Sampling and the Metropolis Algorithm (2005)

    Google Scholar 

  5. Bayati, M., Montanari, A., Saberi, A.: (work in progress, 2007)

    Google Scholar 

  6. Bayati, M., Kim, J.H., Saberi, A.: A Sequential Algorithm for Generating Random Graphs (2007), Extended Version, available at http://arxiv.org/abs/cs/0702124

  7. Bender, E.A., Canfield, E.R.: The asymptotic number of labeled graphs with given degree sequence. J. Combinatorial Theory Ser.A 24(3), 296–307 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bezáková, I., Bhatnagar, N., Vigoda, E.: Sampling Binary Contingency tables with a Greedy Start. In: SODA (2006)

    Google Scholar 

  9. Bezáková, I., Sinclair, A., S̆tefankovič, D., Vigoda, E.: Negative Examples for Sequential Importance Sampling of Binary Contingency Tables, 2006 (preprint)

    Google Scholar 

  10. Blanchet, J.: Efficient Importance Sampling for Counting, 2006 (preprint)

    Google Scholar 

  11. Blitzstein, J., Diaconis, P.: A sequential importance sampling algorithm for generating random graphs with prescribed degrees (submitted)

    Google Scholar 

  12. Bollobás, B.: A probabilistic proof of an asymptotoic forumula for the number of labelled regular graphs. European J. Combin. 1(4), 311–316 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution (preprint)

    Google Scholar 

  14. Bu, T., Towsley, D.: On Distinguishing between Internet Power Law Topology Generator. In: INFOCOM (2002)

    Google Scholar 

  15. Chen, Y., Diaconis, P., Holmes, S., Liu, J.S.: Sequential Monte Carlo methods for statistical analysis of tables. Journal of the American Statistical Association 100, 109–120 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cooper, C., Dyer, M., Greenhill, C.: Sampling regular graphs and peer-to-peer network. Combinatorics, Probability and Computing (to appear)

    Google Scholar 

  17. Chung, F., Lu, L.: Conneted components in random graphs with given expected degree sequence. Ann. Comb. 6(2), 125–145 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Diaconis, P., Gangolli, A.: Rectangular arrays with fixed margins. In: Discrete probability and algorithms (Minneapolis, MN, 1993). IMA Vol. Math. Appl., vol. 72, pp. 15–41. Springer, Heidelberg (1995)

    Google Scholar 

  19. Faloutsos, M., Faloutsos, P., Faloutsos, C.: On Power-law Relationships of the Internet Topology. In: SIGCOM (1999)

    Google Scholar 

  20. Gkantsidis, C., Mihail, M., Zegura, E.: The Markov Chain Simulation Method for Generating Connected Power Law Random Graphs. Alenex (2003)

    Google Scholar 

  21. Jerrum, M., Valiant, L., Vazirani, V.: Random generation of combinatorial structures from a uniform distribution, Theoret. Comput. Sci. 43, 169-188. 73, 1, 91-100 (1986)

    Google Scholar 

  22. Jerrum, M., Sinclair, A.: Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. and Comput. 82(1), 93–133 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jerrum, M., Sinclair, A.: Fast uniform generation of regular graphs. Theoret. Comput. Sci. 73(1), 91–100 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jerrum, M., Sinclair, A., McKay, B.: When is a graphical sequence stable? In: Random graphs vol. 2 (Poznań 1989), pp. 101–115. Wiley-Intersci. Publ. Wiley, New York (1992)

    Google Scholar 

  25. Jerrum, M., Sinclair, A., Vigoda, E.: A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix with Non-Negative Entries. Journal of the ACM 51(4), 671–697 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kannan, R., Tetali, P., Vempala, S.: Simple Markov chain algorithms for generating bipartite graphs and tournaments, (1992) Random Structures and Algorithms 14, 293-308 (1999)

    Google Scholar 

  27. Kim, J.H.: On Brooks’ Theorem for Sparse Graphs. Combi. Prob. & Comp. 4, 97–132 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kim, J.H., Vu, V.H.: Concentration of multivariate polynomials and its applications. Combinatorica 20(3), 417–434 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kim, J.H., Vu, V.H.: Generating Random Regular Graphs. In: STOC, pp. 213–222 (2003)

    Google Scholar 

  30. Kim, J.H., Vu, V.: Sandwiching random graphs. Advances in Mathematics 188, 444–469 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Knuth, D.: Mathematics and computer science: coping with finiteness. Science 194(4271), 1235–1242 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  32. Medina, A., Matta, I., Byers, J.: On the origin of power laws in Internet topologies. ACM Computer Communication Review 30(2), 18–28 (2000)

    Article  Google Scholar 

  33. McKay, B.: Asymptotics for symmetric 0-1 matrices with prescribed row sums. Ars Combinatorica 19A, 15–25 (1985)

    Google Scholar 

  34. McKay, B., Wormald, N.C.: Uniform generation of random regular graphs of moderate degree. J. Algorithms 11(1), 52–67 (1990b)

    Article  MathSciNet  MATH  Google Scholar 

  35. McKay, B., Wormald, N.C.: Asymptotic enumeration by degree sequence of graphs with degrees o(n 1/2). Combinatorica 11(4), 369–382 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Milo, R., Kashtan, N., Itzkovitz, S., Newman, M., Alon, U.: On the uniform generation of random graphs with prescribed degree sequences (2004), http://arxiv.org/PS_cache/cond-mat/pdf/0312/0312028.pdf

  37. Milo, R., ShenOrr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.: Network motifs: Simple building blocks of complex networks. Science 298, 824–827 (2002)

    Article  Google Scholar 

  38. Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Structures and Algorithms 6, 2–3, 161–179 (1995)

    Google Scholar 

  39. Sinclair, A.: Personal communication (2006)

    Google Scholar 

  40. Steger, A., Wormald, N.C.: Generating random regular graphs quickly (English Summary) Random graphs and combinatorial structures (Oberwolfach, 1997). Combin. Probab. Comput. 8(4), 377–396 (1997)

    Article  MATH  Google Scholar 

  41. Tangmunarunkit, H., Govindan, R., Jamin, S., Shenker, S., Willinger, W.: Network Topology Generators: Degree based vs. Structural. ACM SIGCOM (2002)

    Google Scholar 

  42. Wormald, N.C.: Models of random regular graphs. In: Surveys in combinatorics (Canterbury). London Math. Soc. Lecture Notes Ser., vol. 265, pp. 239–298. Cambridge Univ. Press, Cambridge (1999)

    Google Scholar 

  43. Vu, V.H.: Concentration of non-Lipschitz functions and applications, Probabilistic methods in combinatorial optimization. Random Structures Algorithms 20(3), 267–316 (2002)

    Article  Google Scholar 

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Authors and Affiliations

  1. Stanford University, USA

    Mohsen Bayati & Amin Saberi

  2. Yonsei University, South Korea

    Jeong Han Kim

Authors
  1. Mohsen Bayati
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  2. Jeong Han Kim
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  3. Amin Saberi
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Editor information

Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

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Bayati, M., Kim, J.H., Saberi, A. (2007). A Sequential Algorithm for Generating Random Graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_24

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  • DOI: https://doi.org/10.1007/978-3-540-74208-1_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

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