Characterizing optimal sampling of binary contingency tables via the configuration model. Issue 2 (13th February 2012)

• Record Type: Journal Article
• Authors: Blanchet, Jose; Stauffer, Alexandre
• Is Part Of: Random structures & algorithms. Volume 42:Issue 2(2013)
• Page Start: 159
• Page End: 184
• Publication Date: 2012-02-13
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20403
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>A binary contingency table is an <italic>m</italic> × <italic>n</italic> array of binary entries with row sums <bold><italic>r</italic></bold> = (<italic>r</italic><sub>1</sub>, …, <italic>r</italic><sub><italic>m</italic></sub>) and column sums <bold><italic>c</italic></bold> = (<italic>c</italic><sub>1</sub>, …, <italic>c</italic><sub><italic>n</italic></sub>). The configuration model generates a contingency table by considering <italic>r</italic><sub><italic>i</italic></sub> tokens of type 1 for each row <italic>i</italic> and <italic>c</italic><sub><italic>j</italic></sub> tokens of type 2 for each column <italic>j</italic>, and then taking a uniformly random pairing between type‐1 and type‐2 tokens. We give a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as <tex-math notation="LaTeX"><![CDATA[\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}N=\sum_{i=1}^m r_i=\sum_{j=1}^n c_j\end{align*} \end{document}]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg1s17r0km" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> goes to <italic>∞</italic>. Our finding shows surprising differences from recent results for binary <italic>symmetric</italic> contingency tables. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012</p> </abstract>