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August 2011 Random graphs with a given degree sequence
Sourav Chatterjee, Persi Diaconis, Allan Sly
Ann. Appl. Probab. 21(4): 1400-1435 (August 2011). DOI: 10.1214/10-AAP728


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.


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Sourav Chatterjee. Persi Diaconis. Allan Sly. "Random graphs with a given degree sequence." Ann. Appl. Probab. 21 (4) 1400 - 1435, August 2011.


Published: August 2011
First available in Project Euclid: 8 August 2011

zbMATH: 1234.05206
MathSciNet: MR2857452
Digital Object Identifier: 10.1214/10-AAP728

Primary: 05A16 , 05C07 , 05C30 , 52B55 , 60F05 , 62F10 , 62F12

Keywords: degree sequence , Erdős–Gallai criterion , graph limit , random graph , threshold graphs

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.21 • No. 4 • August 2011
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