The Annals of Applied Probability

Random graphs with a given degree sequence

Sourav Chatterjee, Persi Diaconis, and Allan Sly

Full-text: Open access


Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have graph limits in the sense of Lovász and Szegedy with identifiable limits. This allows simple determination of other features such as the number of triangles. The argument proceeds by studying a natural exponential model having the degree sequence as a sufficient statistic. The maximum likelihood estimate (MLE) of the parameters is shown to be unique and consistent with high probability. Thus n parameters can be consistently estimated based on a sample of size one. A fast, provably convergent, algorithm for the MLE is derived. These ingredients combine to prove the graph limit theorem. Along the way, a continuous version of the Erdős–Gallai characterization of degree sequences is derived.

Article information

Ann. Appl. Probab., Volume 21, Number 4 (2011), 1400-1435.

First available in Project Euclid: 8 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A16: Asymptotic enumeration 05C07: Vertex degrees [See also 05E30] 05C30: Enumeration in graph theory 52B55: Computational aspects related to convexity {For computational geometry and algorithms, see 68Q25, 68U05; for numerical algorithms, see 65Yxx} [See also 68Uxx] 60F05: Central limit and other weak theorems 62F10: Point estimation 62F12: Asymptotic properties of estimators

Random graph degree sequence Erdős–Gallai criterion threshold graphs graph limit


Chatterjee, Sourav; Diaconis, Persi; Sly, Allan. Random graphs with a given degree sequence. Ann. Appl. Probab. 21 (2011), no. 4, 1400--1435. doi:10.1214/10-AAP728.

Export citation


  • [1] Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 581–598.
  • [2] Austin, T. (2008). On exchangeable random variables and the statistics of large graphs and hypergraphs. Probab. Surv. 5 80–145.
  • [3] Austin, T. and Tao, T. (2010). Testability and repair of hereditary hypergraph properties. Random Structures Algorithms 36 373–463.
  • [4] Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, Chichester.
  • [5] Barvinok, A. (2010). What does a random contingency table look like? Combin. Probab. Comput. 19 517–539.
  • [6] Barvinok, A. (2010). On the number of matrices and a random matrix with prescribed row and column sums and 0–1 entries. Adv. Math. 224 316–339.
  • [7] Barvinok, A. and Hartigan, J. A. (2009). An asymptotic formula for the number of nonnegative integer matrices with prescibed row and column sums. Preprint. Available at
  • [8] Barvinok, A. and Hartigan, J. A. (2009). Maximum entropy Edgeworth estimates of volumes of polytopes. Preprint. Available at
  • [9] Barvinok, A. and Hartigan, J. A. (2010). Maximum entropy Gaussian approximations for the number of integer points and volumes of polytopes. Adv. in Appl. Math. 45 252–289.
  • [10] Barvinok, A. and Hartigan, J. A. (2010). The number of graphs and a random graph with a given degree sequence. Preprint. Available at
  • [11] Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. MIT Press, Cambridge, MA.
  • [12] Blitzstein, J. and Diaconis, P. (2009). A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Preprint. Available at
  • [13] Borgs, C., Chayes, J., Lovász, L., Sós, V. T. and Vesztergombi, K. (2006). Counting graph homomorphisms. In Topics in Discrete Mathematics. Algorithms Combin. 26 315–371. Springer, Berlin.
  • [14] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
  • [15] Borgs, C., Chayes, J., Lovász, L., Sós, V. T. and Vesztergombi, K. (2007). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Preprint. Available at
  • [16] Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series 9. IMS, Hayward, CA.
  • [17] Diaconis, P. and Freedman, D. (1984). Partial exchangeability and sufficiency. In Statistics: Applications and New Directions (Calcutta, 1981) 205–236. Indian Statist. Inst., Calcutta.
  • [18] Diaconis, P., Holmes, S. and Janson, S. (2008). Threshold graph limits and random threshold graphs. Internet Math. 5 267–320 (2009).
  • [19] Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33–61.
  • [20] Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families. Ann. Statist. 7 269–281.
  • [21] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5 17–61.
  • [22] Erdős, P. and Gallai, T. (1960). Graphen mit punkten vorgeschriebenen grades. Mat. Lapok 11 264–274.
  • [23] Gale, D. (1957). A theorem on flows in networks. Pacific J. Math. 7 1073–1082.
  • [24] Gutiérrez-Peña, E. and Smith, A. F. M. (1995). Conjugate parameterizations for natural exponential families. J. Amer. Statist. Assoc. 90 1347–1356.
  • [25] Gutiérrez-Peña, E. and Smith, A. F. M. (1996). Erratum: “Conjugate parameterizations for natural exponential families.” J. Amer. Statist. Assoc. 91 1757.
  • [26] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • [27] Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. J. Amer. Statist. Assoc. 76 33–65.
  • [28] Hoover, D. N. (1982). Row-column exchangeability and a generalized model for probability. In Exchangeability in Probability and Statistics (Rome, 1981) 281–291. North-Holland, Amsterdam.
  • [29] Hunter, D. R. (2004). MM algorithms for generalized Bradley–Terry models. Ann. Statist. 32 384–406.
  • [30] Jackson, M. O. (2008). Social and Economic Networks. Princeton Univ. Press, Princeton, NJ.
  • [31] Kolaczyk, E. D. (2009). Statistical Analysis of Network Data: Methods and Models. Springer, New York.
  • [32] Lauritzen, S. L. (1988). Extremal Families and Systems of Sufficient Statistics. Lecture Notes in Statistics 49. Springer, New York.
  • [33] Li, L., Alderson, D., Doyle, J. C. and Willinger, W. (2005). Towards a theory of scale-free graphs: Definition, properties, and implications. Internet Math. 2 431–523.
  • [34] Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933–957.
  • [35] Mahadev, N. V. R. and Peled, U. N. (1995). Threshold Graphs and Related Topics. Annals of Discrete Mathematics 56. North-Holland, Amsterdam.
  • [36] McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989) (J. Siemons, ed.). London Mathematical Society Lecture Note Series 141 148–188. Cambridge Univ. Press, Cambridge.
  • [37] McKay, B. D. (2010). Subgraphs of dense random graphs with specified degrees. Preprint. Available at
  • [38] McKay, B. D. and Wormald, N. C. (1990). Asymptotic enumeration by degree sequence of graphs of high degree. European J. Combin. 11 565–580.
  • [39] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161–179.
  • [40] Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295–305.
  • [41] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256 (electronic).
  • [42] Newman, M. E. J., Barabasi, A.-L. and Watts, D. J. (eds.) (2006). The Structure and Dynamics of Networks. Princeton Studies in Complexity. Princeton Univ. Press, Princeton, NJ.
  • [43] Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Phys. Rev. E (3) 70 066117, 13.
  • [44] Portnoy, S. (1984). Asymptotic behavior of M-estimators of p regression parameters when p2/n is large. I. Consistency. Ann. Statist. 12 1298–1309.
  • [45] Portnoy, S. (1985). Asymptotic behavior of M estimators of p regression parameters when p2/n is large. II. Normal approximation. Ann. Statist. 13 1403–1417.
  • [46] Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity. Ann. Statist. 16 356–366.
  • [47] Portnoy, S. (1991). Correction: “Asymptotic behavior of M estimators of p regression parameters when p2/n is large. II. Normal approximation.” Ann. Statist. 19 2282.
  • [48] Robins, G., Snijders, T., Wang, P., Handcock, M. and Pattison, P. (2007). Recent developments in exponential random graph (p) models for social networks. Social Networks 29 192–215.
  • [49] Ryser, H. J. (1957). Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9 371–377.
  • [50] Simons, G. and Yao, Y.-C. (1999). Asymptotics when the number of parameters tends to infinity in the Bradley–Terry model for paired comparisons. Ann. Statist. 27 1041–1060.
  • [51] Tsourakakis, C. (2008). Fast counting of triangles in large real networks: Algorithms and laws. In Proc. of ICDM 2008 608–617. IEEE Computer Society, Los Alamitos, CA.
  • [52] Wainwright, M. J. and Jordan, M. I. (2008). Graphical models, exponential families and variational inference. Foundations and Trends in Machine Learning 1 1–305.
  • [53] Willinger, W., Alderson, D. and Doyle, J. C. (2009). Mathematics and the Internet: A source of enormous confusion and great potential. Notices Amer. Math. Soc. 56 586–599.
  • [54] Wormald, N. C. (1999). Models of random regular graphs. In Surveys in Combinatorics, 1999 (Canterbury). London Mathematical Society Lecture Note Series 267 239–298. Cambridge Univ. Press, Cambridge.