Diameter of the stochastic mean-field model of distance

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 vertices i, j ∈ [n], Janson [18] showed that max i,j∈[n] C _ij/logn converges in probability to 3. We extend these results by showing that max _i,j∈[n] C _ij - 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 Erdos-Rényi random graph in [22].