  A large and sparse random graph with independent exponentially distributed
link weights can be used to model the propagation of messages or diseases in a
network with an unknown connectivity structure. In this article we study an
extended setting where also the nodes of the graph are equipped with
nonnegative random weights which are used to model the effect of boundary
delays across paths in the network. Our main results provide approximative
formulas for typical first passage times, typical flooding times, and maximum
flooding times in the extended setting, over a time scale logarithmic with
respect to the network size.
