Efficient and exact sampling of simple graphs with given arbitrary degree sequence

TL;DR

This paper introduces a fast, exact sampling algorithm for generating independent simple graphs with any specified degree sequence, overcoming limitations of previous methods and enabling accurate network analysis.

Contribution

The authors present a novel polynomial-time algorithm that produces independent graph samples with arbitrary degree sequences, with controlled sampling weights and no rejections.

Findings

Algorithm generates statistically independent samples efficiently.

Samples can be weighted to match desired distributions.

Applicable to power-law and binomial degree sequences.

Abstract

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large N, and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.