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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| Veröffentlicht in: | Annals of combinatorics Jg. 29; H. 2; S. 227 - 272 |
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| 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 $$d_\text {max}$$ 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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| AbstractList | 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
is asymptotically dominated by
, where
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
(
). 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 $$d_\text {max}$$ dmax is asymptotically dominated by $$m^{1/4}$$ 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). 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 ). 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 dmax is asymptotically dominated by 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). 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 max is asymptotically dominated by 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).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 max is asymptotically dominated by 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). |
| Author | Kryven, Ivan van Ieperen, Femke |
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| Cites_doi | 10.1109/FOCS.2019.00084 10.1016/j.ejc.2021.103421 10.37236/9652 10.1016/j.tcs.2017.11.010 10.1002/jgt.20598 10.1016/S0195-6698(80)80030-8 10.1017/S0963548399003867 10.1016/0097-3165(78)90059-6 10.1002/(SICI)1098-2418(199907)14:4<293::AID-RSA1>3.0.CO;2-G 10.1038/ncomms12031 10.1002/rsa.20911 10.1007/s00453-009-9340-1 10.1109/INFCOM.2003.1209234 10.1002/rsa.21105 10.1007/978-3-642-16926-7_21 10.1088/1751-8113/43/19/195002 10.1016/j.dam.2014.10.012 10.1016/j.disc.2021.112566 10.1103/PhysRevE.90.052109 10.1002/rsa.10032 10.1007/978-3-540-31955-9_9 10.1016/j.dam.2020.12.004 10.1137/100799733 10.1371/journal.pone.0201995 10.2140/pjm.1960.10.831 10.37236/1926 10.37236/721 10.1007/s00440-022-01172-7 10.1016/j.jcta.2007.03.009 10.1017/9781009036214.005 10.1080/01621459.2012.758587 10.1080/15427951.2010.557277 10.1088/1367-2630/17/8/083052 10.1145/2488608.2488693 |
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| Keywords | Randomised approximation algorithms Random graphs Directed graphs |
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| SubjectTerms | Algorithms Codes Combinatorial analysis Estimates Expected values Graph theory Graphs Matching Original Paper Random variables Sampling |
| Title | Sequential Stub Matching for Asymptotically Uniform Generation of Directed Graphs with a Given Degree Sequence |
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