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Diameter of the Stochastic Mean-Field Model of Distance

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

Published online by Cambridge University Press:  07 August 2017

SHANKAR BHAMIDI  and
REMCO VAN DER HOFSTAD
SHANKAR BHAMIDI
Affiliation:
Department of Statistics, University of North Carolina, Chapel Hill, NC 27599, USA (e-mail: bhamidi@email.unc.edu)
REMCO VAN DER HOFSTAD
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: rhofstad@win.tue.nl)
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Abstract

We consider the complete graph 𝜅n on n vertices with exponential mean n edge lengths. Writing Cij for the weight of the smallest-weight path between vertices i, j ∈ [n], Janson [18] showed that maxi,j∈[n]Cij/logn converges in probability to 3. We extend these results by showing that maxi,j∈[n]Cij − 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centred graph diameter of the barely supercritical Erdős–Rényi random graph in [22].

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 90B15: Network models, stochastic
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 6 , November 2017 , pp. 797 - 825
Copyright
Copyright © Cambridge University Press 2017 

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