In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.
"The diameter of weighted random graphs." Ann. Appl. Probab. 25 (3) 1686 - 1727, June 2015. https://doi.org/10.1214/14-AAP1034