The scaling limits of several families of random graphs are shown to exist by using the idea of Gromov-Hausdorff convergence to certain random metric spaces.
For instance, uniformly chosen triangulations of the sphere with $n$ faces endowed with the graph distance have been proved to converge (in the Gromov-Hausdorff sense) after rescaling distances by $n^{-1/4}$ to a particular random metric space called the Brownian map. See the references in this earlier answer of mine.