A generalized Pólya's urn with graph based interactions. Issue 4 (21st November 2013)

• Record Type: Journal Article
• Authors: Benaïm, Michel; Benjamini, Itai; Chen, Jun; Lima, Yuri
• Is Part Of: Random structures & algorithms. Volume 46:Issue 4(2015)
• Page Start: 614
• Page End: 634
• Publication Date: 2013-11-21
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20523
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Given a finite connected graph <italic>G</italic>, place a bin at each vertex. Two bins are called a pair if they share an edge of <italic>G</italic>. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7rtmq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20523:rsa20523-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>α</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. We characterize the limiting behavior of the proportion of balls in the bins.</p> <p>The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7rtjp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20523:rsa20523-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>α</mml:mo><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> then there is a single point <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7rtf4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20523:rsa20523-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>, </mml:mo><mml:mo>α</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> with non‐zero entries such that the proportion converges to <italic>v</italic> almost surely.</p> <p>A special case is when <italic>G</italic> is regular and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjmx7rt92" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20523:rsa20523-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>α</mml:mo><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. We show e.g. that if <italic>G</italic> is non‐bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 614–634, 2015</p> </abstract>