Stochastic Systems

Directed random graphs with given degree distributions

Ningyuan Chen and Mariana Olvera-Cravioto

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Given two distributions $F$ and $G$ on the nonnegative integers we propose an algorithm to construct in- and out-degree sequences from samples of i.i.d. observations from $F$ and $G$, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that $F$ and $G$ have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in- and out-degree distributions are (approximately) $F$ and $G$ for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.

Article information

Stoch. Syst., Volume 3, Number 1 (2013), 147-186.

First available in Project Euclid: 24 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability

Directed random graphs simple graphs configuration model prescribed degree distributions


Chen, Ningyuan; Olvera-Cravioto, Mariana. Directed random graphs with given degree distributions. Stoch. Syst. 3 (2013), no. 1, 147--186. doi:10.1214/12-SSY076.

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  • [1] R. Arratia and T.M. Liggett. How likely is an i.i.d. degree sequence to be graphical? Ann. Appl. Probab., 15(1B):652–670, 2005.
  • [2] E.A. Bender and E.R. Canfield. The asymptotic number of labeled graphs with given degree sequences. J. Comb. Theory A, 24(3):296–307, 1978.
  • [3] C. Berge. Graphs and hypergraphs, volume 6. Elsevier, 1976.
  • [4] N.H. Bingham, C.M. Goldie, and J.L. Teugels. Regular variation. Encyclopedia of mathematics and its applications. Cambridge University Press, 1987.
  • [5] J. Blitzstein and P. Diaconis. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Mathematics, 6(4):489–522, 2011.
  • [6] B. Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Combin., 1:311–316, 1980.
  • [7] B. Bollobás. Random graphs, volume 73 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 2001.
  • [8] T. Britton, M. Deijfen, and A. Martin-Löf. Generating simple random graphs with prescribed degree distribution. J. Stat. Phys., 124(6):1377–1397, 2006.
  • [9] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener. Graph structure in the web. Comput. Netw., 33:309–320, 2000.
  • [10] F. Chung and L. Lu. The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci., 99:15879–15882, 2002.
  • [11] F. Chung and L. Lu. Connected components in random graphs with given degree sequences. Ann. Comb., 6:125–145, 2002.
  • [12] P. Erdös and T. Gallai. Graphs with given degree of vertices. Mat. Lapok., 11:264–274, 1960.
  • [13] P. L. Erdös, I. Miklós, and Z. Toroczkai. A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs. Electron J. Comb., 17:1–10, 2010.
  • [14] P.P. Fiziev. Uniform generation of random directed graphs with prescribed degree sequence. Master’s thesis, Max Planck Institute, Germany, March 2006.
  • [15] J. Kleinberg, R. Kumar, R. Raghavan, S. Rajagopalan, and A. Tomkins. Proceedings of the International Conference on Combinatorics and Computing, volume 1627 of Lecture Notes in Computer Science, chapter The Web as a Graph: Measurements, Models, and Methods, pages 1–17. Springer-Verlag, 1999.
  • [16] P.L. Krapivsky and S. Redner. A statistical physics perspective on web growth. Comput. Netw., 39:261–276, 2002.
  • [17] P.L. Krapivsky, G.J. Rodgers, and S. Redner. Degree distributions of growing networks. Phys. Rev. Lett., 86(23):5401–5404, 2001.
  • [18] M.D. LaMar. Algorithms for realizing degree sequences of directed graphs. arXiv:0906.0343, pages 1–35, 2010.
  • [19] B.D. McKay and N.C. Wormald. Asymptotic enumeration by degree sequence of graphs of high degree. Eur. J. Combin., 11:565–580, 1990.
  • [20] B.D. McKay and N.C. Wormald. Uniform generation of random regular graphs of moderate degree. J. Algorithms, 11(1):52–67, 1990.
  • [21] B.D. McKay and N.C. Wormald. Asymptotic enumeration by degree sequence of graphs with degrees $o(n^{1/2})$. Combinatorica, 11:369–382, 1991.
  • [22] M. Molloy and B. Reed. A critical point for random graphs with a given degree sequence. Random Struct. Alg., 6(2-3):161–180, 1995.
  • [23] M.E.J. Newman, S.H. Strogatz, and D.J. Watts. Random graphs with arbitrary degree distributions and their applications. Phys. Rev., 64(2):1–17, 2001.
  • [24] R.C. Read. The enumeration of locally restricted graphs (II). J. Lond. Math. Soc., 35:344–351, 1960.
  • [25] R. van der Hofstad. Random graphs and complex networks., 2012.
  • [26] R. van der Hofstad, G. Hooghiemstra, and P. Van Mieghem. Distances in random graphs with finite variance degrees. Random Struct. Alg., 27:76–123, 2005.
  • [27] N.C. Wormald. Some problems in the enumeration of labelled graphs. PhD thesis, Newcastle University, 1978.
  • [28] N.C. Wormald. Models of random regular graphs. London Mathematical Society Lecture Notes Series, pages 239–298, 1999.