# Sequential Stub Matching for Asymptotically Uniform Generation of Directed Graphs with a Given Degree Sequence

## TL;DR

This paper introduces a method to generate directed graphs with a specific structure while ensuring uniform randomness.

## Contribution

The paper proves that sequential stub matching achieves asymptotically uniform sampling of directed graphs under certain conditions.

## Key findings

- Sequential stub matching can sample simple digraphs with asymptotically equal probability.
- Uniform sampling is achieved when the maximum degree is dominated by m^(1/4).
- The algorithm can be implemented with a linear expected runtime of O(m).

## Abstract

We discuss sequential stub matching for directed graphs and show that this process can be used to sample simple digraphs with asymptotically equal probability. The process starts with an empty edge set and repeatedly adds edges to it with a certain state-dependent bias until the desired degree sequence is fulfilled, whilst avoiding placing a double edge or self-loop. We show that uniform sampling is achieved in the sparse regime when the maximum degree \documentclass[12pt]{minimal}
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\begin{document}$$d_\text {max}$$\end{document}dmax is asymptotically dominated by \documentclass[12pt]{minimal}
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\begin{document}$$m^{1/4}$$\end{document}m1/4, where m is the number of edges. The proof is based on deriving various combinatorial estimates related to the number of digraphs with a given degree sequence and controlling concentration of these estimates in large digraphs. This suggests that sequential stub matching can be viewed as a practical algorithm for almost uniform sampling of digraphs. We show that this algorithm can be implemented to feature a linear expected runtime O(m).