Random graphs with a given degree sequence

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 Lov\'{a}sz 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 Erd\H{o}s--Gallai characterization of degree sequences is derived.

[1]  D. Gale.  A theorem on flows in networks , 1957 .

[2]  H. Ryser.  Combinatorial Properties of Matrices of Zeros and Ones , 1957, Canadian Journal of Mathematics - Journal Canadien de Mathematiques.

[3]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[4]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[5]  P. Diaconis,et al.  Conjugate Priors for Exponential Families , 1979 .

[6]  J. F. C. Kingman,et al.  Information and Exponential Families in Statistical Theory , 1980 .

[7]  D. Aldous.  Representations for partially exchangeable arrays of random variables , 1981 .

[8]  P. Holland,et al.  An Exponential Family of Probability Distributions for Directed Graphs , 1981 .

[9]  S. Portnoy.  Asymptotic Behavior of $M$-Estimators of $p$ Regression Parameters when $p^2/n$ is Large. I. Consistency , 1984 .

[10]  S. Portnoy.  Asymptotic behavior of M-estimators of p regression parameters when p , 1985 .

[11]  B. Bollobás.  The evolution of random graphs , 1984 .

[12]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[13]  L. Brown.  Fundamentals of statistical exponential families: with applications in statistical decision theory , 1986 .

[14]  S. Portnoy.  Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity , 1988 .

[15]  S. Lauritzen.  Extremal Families and Systems of Sufficient Statistics , 1988 .

[16]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[17]  Brendan D. McKay,et al.  Asymptotic Enumeration by Degree Sequence of Graphs of High Degree , 1990, Eur. J. Comb..

[18]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[19]  Adrian F. M. Smith,et al.  Conjugate Parameterizations for Natural Exponential Families , 1995 .

[20]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[21]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

[22]  N. Wormald,et al.  Models of the , 2010 .

[23]  Yi-Ching Yao,et al.  Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons , 1999 .

[24]  D. Hunter.  MM algorithms for generalized Bradley-Terry models , 2003 .

[25]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[26]  M. Newman,et al.  Statistical mechanics of networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Walter Willinger,et al.  Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications , 2005, Internet Math..

[28]  László Lovász,et al.  Limits of dense graph sequences , 2004, J. Comb. Theory B.

[29]  P. Valtr,et al.  Topics in Discrete Mathematics , 2006 .

[30]  V. Sós,et al.  Counting Graph Homomorphisms , 2006 .

[31]  Duncan J. Watts,et al.  The Structure and Dynamics of Networks: (Princeton Studies in Complexity) , 2006 .

[32]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[33]  Peng Wang,et al.  Recent developments in exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[34]  V. Sós,et al.  Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing , 2007, math/0702004.

[35]  P. Diaconis,et al.  Graph limits and exchangeable random graphs , 2007, 0712.2749.

[36]  Tim Austin.  On exchangeable random variables and the statistics of large graphs and hypergraphs , 2008, 0801.1698.

[37]  Svante Janson,et al.  Threshold Graph Limits and Random Threshold Graphs , 2008, Internet Math..

[38]  A. Barvinok.  On the number of matrices and a random matrix with prescribed row and column sums and 0–1 entries☆ , 2008, 0806.1480.

[39]  Charalampos E. Tsourakakis.  Fast Counting of Triangles in Large Real Networks : Algorithms and Laws , 2008 .

[40]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[41]  Eric D. Kolaczyk,et al.  Statistical Analysis of Network Data: Methods and Models , 2009 .

[42]  Alexander Barvinok,et al.  An asymptotic formula for the number of non-negative integer matrices with prescribed row and column sums , 2009 .

[43]  J. Hartigan,et al.  Maximum Entropy Edgeworth Estimates of Volumes of Polytopes , 2009 .

[44]  Alexander Barvinok,et al.  Maximum entropy Gaussian approximations for the number of integer points and volumes of polytopes , 2010, Adv. Appl. Math..

[45]  Terence Tao,et al.  Testability and repair of hereditary hypergraph properties , 2008, Random Struct. Algorithms.

[46]  Alexander I. Barvinok,et al.  What Does a Random Contingency Table Look Like? , 2008, Combinatorics, Probability and Computing.

[47]  N. Wormald.  Models of random regular graphs , 2010 .

[48]  Brendan D. McKay,et al.  Subgraphs of Dense Random Graphs with Specified Degrees , 2010, Combinatorics, Probability and Computing.

[49]  Walter Willinger,et al.  Mathematics and the Internet: A Source of Enormous Confusion and Great Potential , 2009, The Best Writing on Mathematics 2010.

[50]  Persi Diaconis,et al.  A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees , 2011, Internet Math..

[51]  V. Sós,et al.  Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics , 2012 .

[52]  Alexander I. Barvinok,et al.  The number of graphs and a random graph with a given degree sequence , 2010, Random Struct. Algorithms.