Random graphs with a given degree sequence

TL;DR

This paper investigates the properties of large graphs with fixed degree sequences, establishing their convergence to graph limits, and introduces methods for statistical estimation and analysis of such graphs.

Contribution

It provides a framework for understanding the limits of graphs with given degree sequences and introduces a consistent MLE and an efficient algorithm for parameter estimation.

Findings

Graphs with fixed degree sequences have identifiable graph limits.

A unique and consistent MLE for the degree sequence parameters is established.

A new continuous Erdős–Gallai characterization of degree sequences is derived.

Abstract

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.