Half-Graphs, Other Non-stable Degree Sequences, and the Switch Markov Chain

Erdős, Péter L.; Győri, Ervin; Mezei, Tamás Róbert; Miklós, István; Soltész, Daniel

doi:10.37236/9652

Cited by 2 publications

(1 citation statement)

References 26 publications

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“…Hence the class {k(n) | n ≥ 2} is not P-stable. Erdős et al [40] described more general classes of degree sequences with these properties. So Pstability is not a necessary condition for the switch chain to be efficient.…”

Section: Strong Stability Amanatidis and Kleermentioning

confidence: 99%

Generating graphs randomly

Greenhill¹

2022

Preprint

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Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which are similar in some way. One way to do this is to take a sample of several random graphs from the family, to gather information about what is "typical". Hence there is a need for algorithms which can generate graphs uniformly (or approximately uniformly) at random from the given family. Since a large sample may be required, the algorithm should also be computationally efficient.Rigorous analysis of such algorithms is often challenging, involving both combinatorial and probabilistic arguments. We will focus mainly on the set of all simple graphs with a particular degree sequence, and describe several different algorithms for sampling graphs from this family uniformly, or almost uniformly.

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Section: Strong Stability Amanatidis and Kleermentioning

confidence: 99%

Generating graphs randomly

Greenhill¹

2022

Preprint

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

van Ieperen,

Kryven

2024

Ann. Comb.

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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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Half-Graphs, Other Non-stable Degree Sequences, and the Switch Markov Chain

Cited by 2 publications

References 26 publications

Generating graphs randomly

Generating graphs randomly

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

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