Sampling random graphs with specified degree sequences

TL;DR

This paper introduces a new method to accurately detect convergence in Markov chain Monte Carlo sampling of random graphs with fixed degree sequences, improving the efficiency of generating null models for network analysis.

Contribution

It presents an algorithm using assortativity and a heuristic for estimating the MCMC gap, along with a convergence detection method based on the Dickey-Fuller test, enhancing sampling accuracy.

Findings

The proposed method outperforms existing convergence tests in accuracy and efficiency.

The algorithm effectively estimates the independence gap in MCMC sampling.

Validated on 509 empirical networks, demonstrating practical applicability.

Abstract

The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network's structure can be explained by its degree structure alone. A Markov chain Monte Carlo (MCMC) algorithm, based on a degree-preserving double-edge swap, provides an asymptotic solution to sample from the configuration model. However, accurately and efficiently detecting this Markov chain's convergence on its stationary distribution remains an unsolved problem. Here, we provide a solution to detect convergence and sample from the configuration model. We develop an algorithm, based on the assortativity of the sampled graphs, for estimating the gap between effectively independent MCMC states, and a computationally efficient gap-estimation heuristic derived from analyzing a corpus of 509 empirical networks. We provide a convergence detection method based on the Dickey-Fuller Generalized Least Squares test, which we show is more accurate and efficient than three alternative Markov chain convergence tests.