Constructing and sampling directed graphs with given degree sequences

TL;DR

This paper introduces a rejection-free algorithm for constructing all possible directed graphs with a given degree sequence, enabling independent and weighted sampling for statistical analysis of complex directed networks.

Contribution

The authors present a novel rejection-free method for generating all realizations of a bi-degree sequence in directed graphs, ensuring independent samples and providing their weights.

Findings

The method guarantees rejection-free, independent sampling of graph realizations.

It allows computation of statistical averages using weighted samples.

The approach improves over existing methods by avoiding unknown mixing times and rejection issues.

Abstract

The interactions between the components of complex networks are often directed. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. As the number of simple labeled graphs with a given degree sequence is typically very large even for short sequences, sampling methods are needed for statistical studies. Currently, there are two main classes of methods that generate samples. One of the existing methods first generates a restricted class of graphs, then uses a Markov Chain Monte-Carlo algorithm based on edge swaps to generate other realizations. As the mixing time of this process is still unknown, the independence of the samples is not well controlled. The other class of methods is based on the Configuration Model that may lead to unacceptably many sample rejections due to self-loops and multiple edges. Here we present an algorithm that can directly construct all possible realizations of a given bi-degree sequence by simple digraphs. Our method is rejection free, guarantees the independence of the constructed samples, and provides their weight. The weights can then be used to compute statistical averages of network observables as if they were obtained from uniformly distributed sampling, or from any other chosen distribution.