  We present a framework to simulate SIR processes on networks using weighted
shortest paths. Our framework maps the SIR dynamics to weights assigned to the
edges of the network, which can be done for Markovian and non-Markovian
processes alike. The weights represent the propagation time between the
adjacent nodes for a particular realization. We simulate the dynamics by
constructing an ensemble of such realizations, which can be done by using a
Markov Chain Monte Carlo method or by direct sampling. The former provides a
runtime advantage when realizations from all possible sources are computed as
the weighted shortest paths can be re-calculated more efficiently. We apply our
framework to three empirical networks and analyze the expected propagation time
between all pairs of nodes. Furthermore, we have employed our framework to
perform efficient source detection and to improve strategies for time-critical
vaccination.
