Uniform random generation of large acyclic digraphs

TL;DR

This paper presents a novel, efficient method for uniformly sampling large acyclic digraphs using recursive enumeration and a hybrid Markov chain, improving over existing approaches in speed and flexibility.

Contribution

It introduces a recursive enumeration-based exact sampling method and a new hybrid Markov chain for faster uniform sampling of large acyclic digraphs, including restricted cases.

Findings

Recursive enumeration provides an exact sampling method avoiding convergence issues.

The hybrid Markov chain converges faster than existing methods.

The approach enables efficient uniform sampling of large and restricted acyclic digraphs.

Abstract

Directed acyclic graphs are the basic representation of the structure underlying Bayesian networks, which represent multivariate probability distributions. In many practical applications, such as the reverse engineering of gene regulatory networks, not only the estimation of model parameters but the reconstruction of the structure itself is of great interest. As well as for the assessment of different structure learning algorithms in simulation studies, a uniform sample from the space of directed acyclic graphs is required to evaluate the prevalence of certain structural features. Here we analyse how to sample acyclic digraphs uniformly at random through recursive enumeration, an approach previously thought too computationally involved. Based on complexity considerations, we discuss in particular how the enumeration directly provides an exact method, which avoids the convergence issues of the alternative Markov chain methods and is actually computationally much faster. The limiting behaviour of the distribution of acyclic digraphs then allows us to sample arbitrarily large graphs. Building on the ideas of recursive enumeration based sampling we also introduce a novel hybrid Markov chain with much faster convergence than current alternatives while still being easy to adapt to various restrictions. Finally we discuss how to include such restrictions in the combinatorial enumeration and the new hybrid Markov chain method for efficient uniform sampling of the corresponding graphs.