# Diameter of the stochastic mean-field model of distance

### Uittreksel

We consider the complete graph K_n on n vertices with exponential mean n edge lengths. Writing C_{ij} for the weight of the smallest-weight path between vertex i,j \in [n], Janson [17] showed that max_{i,j \in [n]} C_{ij} / log n converges in probability to 3. We extend this results by showing that max_{i,j \in [n]} C_{ij} - 3 log n 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-centered graph diameter of the barely supercritical Erdös-Rényi random graph in [21].
Originele taal-2 Engels Eindhoven Eurandom 27 Gepubliceerd - 2013

### Publicatie series

Naam Report Eurandom 2013013 1389-2355

### Vingerafdruk

Mean-field Model
Stochastic Model
Limiting
Random variable
Converge
Random Structure
Weak Limit
Random Graphs
Complete Graph
Path
Graph in graph theory
Vertex of a graph

### Citeer dit

Bhamidi, S., & Hofstad, van der, R. W. (2013). Diameter of the stochastic mean-field model of distance. (Report Eurandom; Vol. 2013013). Eindhoven: Eurandom.
Bhamidi, S. ; Hofstad, van der, R.W. / Diameter of the stochastic mean-field model of distance. Eindhoven : Eurandom, 2013. 27 blz. (Report Eurandom).
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title = "Diameter of the stochastic mean-field model of distance",
abstract = "We consider the complete graph K_n on n vertices with exponential mean n edge lengths. Writing C_{ij} for the weight of the smallest-weight path between vertex i,j \in [n], Janson [17] showed that max_{i,j \in [n]} C_{ij} / log n converges in probability to 3. We extend this results by showing that max_{i,j \in [n]} C_{ij} - 3 log n 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-centered graph diameter of the barely supercritical Erd{\"o}s-R{\'e}nyi random graph in [21].",
author = "S. Bhamidi and {Hofstad, van der}, R.W.",
year = "2013",
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Bhamidi, S & Hofstad, van der, RW 2013, Diameter of the stochastic mean-field model of distance. Report Eurandom, vol. 2013013, Eurandom, Eindhoven.
Eindhoven : Eurandom, 2013. 27 blz. (Report Eurandom; Vol. 2013013).

TY - BOOK

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AU - Hofstad, van der, R.W.

PY - 2013

Y1 - 2013

N2 - We consider the complete graph K_n on n vertices with exponential mean n edge lengths. Writing C_{ij} for the weight of the smallest-weight path between vertex i,j \in [n], Janson [17] showed that max_{i,j \in [n]} C_{ij} / log n converges in probability to 3. We extend this results by showing that max_{i,j \in [n]} C_{ij} - 3 log n 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-centered graph diameter of the barely supercritical Erdös-Rényi random graph in [21].

AB - We consider the complete graph K_n on n vertices with exponential mean n edge lengths. Writing C_{ij} for the weight of the smallest-weight path between vertex i,j \in [n], Janson [17] showed that max_{i,j \in [n]} C_{ij} / log n converges in probability to 3. We extend this results by showing that max_{i,j \in [n]} C_{ij} - 3 log n 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-centered graph diameter of the barely supercritical Erdös-Rényi random graph in [21].

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Bhamidi S, Hofstad, van der RW. Diameter of the stochastic mean-field model of distance. Eindhoven: Eurandom, 2013. 27 blz. (Report Eurandom).