<abstract xmlns="http://www.w3.org/1999/xhtml">Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, <xref ref-type="bibr" rid="j_ausm-2018-0020_ref_001_w2aab3b7b3b1b6b1ab1ab1Aa">Kim and Saberi (2010)</xref> that has a running time of O(mamax), where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2018-0020_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><msub><mrow><mi>a</mi></mrow><mi>i</mi></msub></mrow></mrow></math><tex-math>$m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}}</tex-math></alternatives></inline-formula> and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by <xref ref-type="bibr" rid="j_ausm-2018-0020_ref_008_w2aab3b7b3b1b6b1ab1ab8Aa">Chen et al. (2005)</xref> does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by <xref ref-type="bibr" rid="j_ausm-2018-0020_ref_006_w2aab3b7b3b1b6b1ab1ab6Aa">Brualdi (1980)</xref> and <xref ref-type="bibr" rid="j_ausm-2018-0020_ref_021_w2aab3b7b3b1b6b1ab1ac21Aa">Kannan et al.(1999)</xref>.</abstract>

Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where m=12∑i=1nai$m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}} and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999).

Rejection sampling of bipartite graphs with given degree sequence

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Rejection sampling of bipartite graphs with given degree sequence"}

Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite...

Rejection sampling of bipartite graphs with given degree sequence

Published Online: Mar 04, 2019

Page range: 249 - 275

Received: Mar 28, 2018

DOI: https://doi.org/10.2478/ausm-2018-0020

Keywords
degree sequence, contraction of a degree sequence, degree sequence bipartition, contraction of a graph, deletion of a graph, ecological occurrence matrix

© 2018 Koko K. Kayibi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Rejection sampling of bipartite graphs with given degree sequence

Published Online: Mar 04, 2019

Page range: 249 - 275

Received: Mar 28, 2018

DOI: https://doi.org/10.2478/ausm-2018-0020

Keywordsdegree sequence, contraction of a degree sequence, degree sequence bipartition, contraction of a graph, deletion of a graph, ecological occurrence matrix

© 2018 Koko K. Kayibi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
degree sequence, contraction of a degree sequence, degree sequence bipartition, contraction of a graph, deletion of a graph, ecological occurrence matrix