Random graphs with a giüen degree sequence

Annals of Applied Probability

Volumn 21, Issue 4, 2011, Pages 1400-1435

  Chatterjee, Sourav ^a   Diaconis, Persi ^b   Sly, Allan ^c  

^a NEW YORK UNIVERSITY   (United States)

^b STANFORD UNIVERSITY   (United States)

^c MICROSOFT RESEARCH   (United States)

Author keywords

Degree sequence; Erd s Gallai criterion; Graph limit; Random graph; Threshold graphs

Indexed keywords

EID: 80051625575     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AAP728     Document Type: Article

References (54)

1. ALDOUS, D. J. (1981). Representations for partially exchangeable arrays of random üariables. J. Multiüariate Anal. 11 581-598. MR0637937
2. AUSTIN, T. (2008). On exchangeable random üariables and the statistics of large graphs and hypergraphs. Probab. Surü. 5 80-145. MR2426176
3. AUSTIN, T. and TAO, T. (2010). Testability and repair of hereditary hypergraph properties. Random Structures Algorithms 36 373-463. MR2666763
4. BARNDORFF-NIELSEN, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, Chichester. MR0489333
5. BARVINOK, A. (2010). What does a random contingency table look like? Combin. Probab. Comput. 19 517-539. MR2647491
6. BARVINOK, A. (2010). On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries. Adü. Math. 224 316-339. MR2600999
7. BARVINOK, A. and HARTIGAN, J. A. (2009). An asymptotic formula for the number of nonnegatiüe integer matrices with prescibed row and column sums. Preprint. Available at http://arxiu.org/abs/0910.2477.
8. BARVINOK, A. and HARTIGAN, J. A. (2009). Maximum entropy Edgeworth estimates of üolumes of polytopes. Preprint. Available at http://arxiv.org/abs/0910.2497.
9. BARVINOK, A. andHARTIGAN, J. A. (2010).Maximum entropy Gaussian approximations for the number of integer points and üolumes of polytopes. Adü. in Appl. Math. 45 252-289. MR2646125
10. BARVINOK, A. and HARTIGAN, J. A. (2010). The number of graphs and a random graph with a giüen degree sequence. Preprint. Available at http://arxiv.org/abs/1003.0356.
11. BISHOP, Y. M. M., FIENBERG, S. E. and HOLLAND, P. W. (1975). Discrete Multiüariate 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 http://www. people.fas.haruard.edu/blitz/ BlitzsteinDiaconisGraphAlgorithm.pdf.
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. MR2249277
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. MR2455626
15. BORGS, C., CHAYES, J., LOVÁSZ, L., SÓS, V. T. and VESZTERGOMBI, K. (2007). Conüergent sequences of dense graphs II. Multiway cuts and statistical physics. Preprint. Available at http://research.microsoft. com/en-us/um/people/borgs/papers/conright.pdf.
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. MR0882001
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. MR0786142
18. DIACONIS, P., HOLMES, S. and JANSON, S. (2008). Threshold graph limits and random threshold graphs. Internet Math. 5 267-320 (2009). MR2573956
19. DIACONIS, P. and JANSON, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33-61. MR2463439
20. DIACONIS, P. and YLVISAKER, D. (1979). Conjugate priors for exponential families. Ann. Statist. 7 269-281. MR0520238
21. ERDÖ S, P. and RéNYI, A. (1960). On the eüolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5 17-61. MR0125031
22. ERDÖ S, P. and GALLAI, T. (1960). Graphen mit punkten üorgeschriebenen grades. Mat. Lapok 11 264-274.
23. GALE, D. (1957). A theorem on flows in networks. Pacific J. Math. 7 1073-1082. MR0091855
24. GUTIéRREZ-PEñA, E. and SMITH, A. F. M. (1995). Conjugate parameterizations for natural exponential families. J. Amer. Statist. Assoc. 90 1347-1356. MR1379477 (Pubitemid 126093148)
25. GUTIéRREZ-PEñA, E. and SMITH, A. F. M. (1996). Erratum: "Conjugate parameterizations for natural exponential families." J. Amer. Statist. Assoc. 91 1757. MR1439120
26. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random üariables. J. Amer. Statist. Assoc. 58 13-30. MR0144363
27. HOLLAND, P. W. and LEINHARDT, S. (1981). An exponential family of probability distributions for directed graphs. J. Amer. Statist. Assoc. 76 33-65. MR0608176
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. MR0675982
29. HUNTER, D. R. (2004). MM algorithms for generalized Bradley-Terry models. Ann. Statist. 32 384-406. MR2051012 (Pubitemid 41449316)
30. JACKSON, M. O. (2008). Social and Economic Networks. Princeton Uniü. Press, Princeton, NJ. MR2435744
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. MR0971253
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. MR2241756
34. LOVÁSZ, L. and SZEGEDY, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933-957. MR2274085 (Pubitemid 44858105)
35. MAHADEV, N. V. R. and PELED, U. N. (1995). Threshold Graphs and Related Topics. Annals of Discrete Mathematics 56. North-Holland, Amsterdam. MR1417258
36. MCDIARMID, C. (1989). On the method of bounded differences. In Surüeys in Combinatorics, 1989 (Norwich, 1989) (J. Siemons, ed.). London Mathematical Society Lecture Note Series 141 148-188. Cambridge Uniü. Press, Cambridge. MR1036755
37. MCKAY, B. D. (2010). Subgraphs of dense random graphs with specified degrees. Preprint. Available at http://arxiv.org/abs/1002.3018.
38. MCKAY, B. D. andWORMALD, N. C. (1990). Asymptotic enumeration by degree sequence of graphs of high degree. European J. Combin. 11 565-580. MR1078713
39. MOLLOY, M. and REED, B. (1995). A critical point for random graphs with a giüen degree sequence. Random Structures Algorithms 6 161-179. MR1370952
40. MOLLOY, M. and REED, B. (1998). The size of the giant component of a random graph with a giüen degree sequence. Combin. Probab. Comput. 7 295-305. MR1664335 (Pubitemid 128480814)
41. NEWMAN, M. E. J. (2003). The structure and function of complex networks. SIAM Reü. 45 167-256 (electronic). MR2010377
42. NEWMAN, M. E. J., BARABASI, A.-L. andWATTS, D. J. (EDS.) (2006). The Structure and Dynamics of Networks. Princeton Studies in Complexity. Princeton Uniü. Press, Princeton, NJ. MR2352222
43. PARK, J. and NEWMAN, M. E. J. (2004). Statistical mechanics of networks. Phys. Reü. E (3) 70 066117, 13. MR2133807
44. PORTNOY, S. (1984). Asymptotic behaüior of M-estimators of p regression parameters when p2/n is large. I. Consistency. Ann. Statist. 12 1298-1309. MR0760690
45. PORTNOY, S. (1985). Asymptotic behaüior of M estimators of p regression parameters when p2/n is large. II. Normal approximation. Ann. Statist. 13 1403-1417. MR0811499
46. PORTNOY, S. (1988). Asymptotic behaüior of likelihood methods for exponential families when the number of parameters tends to infinity. Ann. Statist. 16 356-366. MR0924876
47. PORTNOY, S. (1991). Correction: "Asymptotic behaüior of M estimators of p regression parameters when p2/n is large. II. Normal approximation." Ann. Statist. 19 2282. MR1135181
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. MR0087622
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. MR1724040
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 üariational 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. MR2509062
54. WORMALD, N. C. (1999). Models of random regular graphs. In Surüeys in Combinatorics, 1999 (Canterbury). London Mathematical Society Lecture Note Series 267 239-298. Cambridge Uniü. Press, Cambridge. MR1725006