  In this paper, we study the impact of edge weights on distances in diluted
random graphs. We interpret these weights as delays, and take them as i.i.d
exponential random variables. We analyze the weighted flooding time defined as
the minimum time needed to reach all nodes from one uniformly chosen node, and
the weighted diameter corresponding to the largest distance between any pair of
vertices. Under some regularity conditions on the degree sequence of the random
graph, we show that these quantities grow as the logarithm of $n$, when the
size of the graph $n$ tends to infinity. We also derive the exact value for the
prefactors. These allow us to analyze an asynchronous randomized broadcast
algorithm for random regular graphs. Our results show that the asynchronous
version of the algorithm performs better than its synchronized version: in the
large size limit of the graph, it will reach the whole network faster even if
the local dynamics are similar on average.
