We study in this thesis the asymptotic behavior of optimal paths on a random graph model, the configuration model, for which we assign continuous random positive weights on its edges. We start by describing the asymptotic behavior of the diameter and the flooding time on the graph for a set of light-tailed edge-weights, and we prove that it is the largest class of densities for which these precise asymptotic expressions for the diameter/flooding time hold. We then show how the weighted optimal path can be constructed using a particular positive recurrent Markov chain. We finally study, in the last chapter, the noise sensitivity of the model by replacing every edge-weight independently by a new realization of the same distribution, with probability epsilon. We show that the optimal paths between two vertices before and after this modification are asymptotically ''independent'' conditioning on the graph, as the number of vertices tends to infinity.