Annals of Applied Probability

Random graphs with a given degree sequence

Sourav Chatterjee, Persi Diaconis, and Allan Sly

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

First available in Project Euclid: 8 August 2011

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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.

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