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Joint Distribution of Distances in Large Random Regular Networks

Part of: Combinatorial probability Graph theory Operations research and management science

Published online by Cambridge University Press:  30 January 2018

Justin Salez
Justin Salez*
Affiliation:
University of California, Berkeley
*
∗ Current address: Université Paris Diderot, LPMA, Site Chevaleret, case 7012, 75205 Paris Cedex 13, France. Email address: [email protected]
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Abstract

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We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

Keywords

Random regular graphdistance matrixfirst passage percolationmultitype Richardson processconfiguration modelbranching process approximation

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 90B15: Network models, stochastic
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 861 - 870
Copyright
© Applied Probability Trust 

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