A Sequential Algorithm for Generating Random Graphs

  • Mohsen Bayati
  • Jeong Han Kim
  • Amin Saberi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4627)


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(md 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(m3n2). 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(n1/2 − τ), our algorithm can generate an asymptotically uniform d-regular graph. Our results are improving the previous bound of d = O(n1/3 − τ) due to Kim and Vu [30] for regular graphs.


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  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley, NewYork (1992)MATHGoogle Scholar
  2. 2.
    Alderson, D., Doyle, J., Willinger, W.: Toward and Optimization-Driven Framework for Designing and Generating Realistic Internet Topologies. HotNets (2002)Google Scholar
  3. 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. 4.
    Bassetti, F., Diaconis, P.: Examples Comparing Importance Sampling and the Metropolis Algorithm (2005)Google Scholar
  5. 5.
    Bayati, M., Montanari, A., Saberi, A.: (work in progress, 2007)Google Scholar
  6. 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bezáková, I., Bhatnagar, N., Vigoda, E.: Sampling Binary Contingency tables with a Greedy Start. In: SODA (2006)Google Scholar
  9. 9.
    Bezáková, I., Sinclair, A., S̆tefankovič, D., Vigoda, E.: Negative Examples for Sequential Importance Sampling of Binary Contingency Tables, 2006 (preprint)Google Scholar
  10. 10.
    Blanchet, J.: Efficient Importance Sampling for Counting, 2006 (preprint)Google Scholar
  11. 11.
    Blitzstein, J., Diaconis, P.: A sequential importance sampling algorithm for generating random graphs with prescribed degrees (submitted)Google Scholar
  12. 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)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution (preprint)Google Scholar
  14. 14.
    Bu, T., Towsley, D.: On Distinguishing between Internet Power Law Topology Generator. In: INFOCOM (2002)Google Scholar
  15. 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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cooper, C., Dyer, M., Greenhill, C.: Sampling regular graphs and peer-to-peer network. Combinatorics, Probability and Computing (to appear)Google Scholar
  17. 17.
    Chung, F., Lu, L.: Conneted components in random graphs with given expected degree sequence. Ann. Comb. 6(2), 125–145 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 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. 19.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On Power-law Relationships of the Internet Topology. In: SIGCOM (1999)Google Scholar
  20. 20.
    Gkantsidis, C., Mihail, M., Zegura, E.: The Markov Chain Simulation Method for Generating Connected Power Law Random Graphs. Alenex (2003)Google Scholar
  21. 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. 22.
    Jerrum, M., Sinclair, A.: Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. and Comput. 82(1), 93–133 (1989)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jerrum, M., Sinclair, A.: Fast uniform generation of regular graphs. Theoret. Comput. Sci. 73(1), 91–100 (1990)MathSciNetCrossRefMATHGoogle Scholar
  24. 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  26. 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. 27.
    Kim, J.H.: On Brooks’ Theorem for Sparse Graphs. Combi. Prob. & Comp. 4, 97–132 (1995)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kim, J.H., Vu, V.H.: Concentration of multivariate polynomials and its applications. Combinatorica 20(3), 417–434 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kim, J.H., Vu, V.H.: Generating Random Regular Graphs. In: STOC, pp. 213–222 (2003)Google Scholar
  30. 30.
    Kim, J.H., Vu, V.: Sandwiching random graphs. Advances in Mathematics 188, 444–469 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Knuth, D.: Mathematics and computer science: coping with finiteness. Science 194(4271), 1235–1242 (1976)MathSciNetCrossRefMATHGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. 33.
    McKay, B.: Asymptotics for symmetric 0-1 matrices with prescribed row sums. Ars Combinatorica 19A, 15–25 (1985)Google Scholar
  34. 34.
    McKay, B., Wormald, N.C.: Uniform generation of random regular graphs of moderate degree. J. Algorithms 11(1), 52–67 (1990b)MathSciNetCrossRefMATHGoogle Scholar
  35. 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)MathSciNetCrossRefMATHGoogle Scholar
  36. 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. 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)CrossRefGoogle Scholar
  38. 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. 39.
    Sinclair, A.: Personal communication (2006)Google Scholar
  40. 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)CrossRefMATHGoogle Scholar
  41. 41.
    Tangmunarunkit, H., Govindan, R., Jamin, S., Shenker, S., Willinger, W.: Network Topology Generators: Degree based vs. Structural. ACM SIGCOM (2002)Google Scholar
  42. 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. 43.
    Vu, V.H.: Concentration of non-Lipschitz functions and applications, Probabilistic methods in combinatorial optimization. Random Structures Algorithms 20(3), 267–316 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Mohsen Bayati
    • 1
  • Jeong Han Kim
    • 2
  • Amin Saberi
    • 1
  1. 1.Stanford UniversityUSA
  2. 2.Yonsei UniversitySouth Korea

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