Unbiased degree-preserving randomisation of directed binary networks

TL;DR

This paper introduces an unbiased, degree-preserving randomisation method for directed binary networks using a Markov chain with acceptance probabilities, enabling uniform sampling and customizable topological features.

Contribution

It develops a novel ergodic Markov chain with detailed balance for directed graphs that preserves degrees and can target arbitrary measures.

Findings

The method produces unbiased randomised directed networks.

It can generate graphs with specified degree-degree correlations.

Numerical tests confirm effectiveness on synthetic and biological networks.

Abstract

Randomising networks using a naive `accept-all' edge-swap algorithm is generally biased. Building on recent results for nondirected graphs, we construct an ergodic detailed balance Markov chain with non-trivial acceptance probabilities for directed graphs, which converges to a strictly uniform measure and is based on edge swaps that conserve all in- and out-degrees. The acceptance probabilities can also be generalized to define Markov chains that target any alternative desired measure on the space of directed graphs, in order to generate graphs with more sophisticated topological features. This is demonstrated by defining a process tailored to the production of directed graphs with specified degree-degree correlation functions. The theory is implemented numerically and tested on synthetic and biological network examples.