In this talk, we study two different diffusion models on the random graphs. In the first part, we consider first passage percolation. We analyze the impact of the edge weights on distances in sparse random graphs. Our main result consists of a precise asymptotic expression for the weighted diameter of sparse random graphs when the edge weights are i.i.d. exponential random variables.  In the second part, we model the propagation of defaults across financial institutions as a cascade process on a network representing their mutual exposures. We derive rigorous asymptotic results for the magnitude of contagion in a large financial network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. We also introduce a criterion for the resilience of a large inhomogeneous random financial network to initial shocks that can be used as a tool for monitoring systemic risk.