Diameter of the Stochastic Mean-Field Model of Distance

Author:

BHAMIDI SHANKAR,VAN DER HOFSTAD REMCO

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].

Publisher

Cambridge University Press (CUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science

Cited by 6 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Long-Range First-Passage Percolation on the Torus;Journal of Statistical Physics;2024-08-27

2. Minimum weight disk triangulations and fillings;Transactions of the American Mathematical Society;2021-02-12

3. Diameter and Stationary Distribution of Random $r$-Out Digraphs;The Electronic Journal of Combinatorics;2020-08-07

4. Long Paths in First Passage Percolation on the Complete Graph II. Global Branching Dynamics;Journal of Statistical Physics;2020-08-06

5. Long paths in first passage percolation on the complete graph I. Local PWIT dynamics;Electronic Journal of Probability;2020-01-01

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