Connectivity of inhomogeneous random graphs1. Issue 3 (26th February 2013)

• Record Type: Journal Article
• Authors: Devroye, Luc; Fraiman, Nicolas
• Is Part Of: Random structures & algorithms. Volume 45:Issue 3(2014)
• Page Start: 408
• Page End: 420
• Publication Date: 2013-02-26
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20490
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for <italic>G</italic>(<italic>n, p</italic>), when <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v95q8g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20490:rsa20490-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>. We draw <italic>n</italic> independent points <italic>X</italic><sub><italic>i</italic></sub> from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (<italic>i, j</italic>) is added with probability <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v95qbk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20490:rsa20490-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>min</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>, </mml:mo><mml:mo>κ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>, </mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v95qwd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20490:rsa20490-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>κ</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 408‐420, 2014</p> </abstract>