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

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 seque...

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Bibliographic Details
Published in:Annals of combinatorics Vol. 29; no. 2; pp. 227 - 272
Main Authors: van Ieperen, Femke, Kryven, Ivan
Format: Journal Article
Language:English
Published: Switzerland Springer Nature B.V 01.06.2025
Springer International Publishing
Subjects:
Algorithms
Codes
Combinatorial analysis
Estimates
Expected values
Graph theory
Graphs
Matching
Original Paper
Random variables
Sampling
Randomised approximation algorithms
Random graphs
Directed graphs
ISSN:0218-0006, 0219-3094
Online Access:Get full text
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Summary: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 $$d_\text {max}$$ d max is asymptotically dominated by $$m^{1/4}$$ m 1 / 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 ).
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Communicated by Frédérique Bassino.
ISSN:0218-0006
0219-3094
DOI:10.1007/s00026-024-00715-0