Mapping spreading dynamics: From time respecting shortest paths to bond percolation

We propose a mapping of spreading dynamics to an ensemble of weighted networks, where edge weights represent propagation time delays. In this mapping, shortest paths in the weighted networks preserve the temporal causality of spreading. Furthermore, for efficient sampling, we construct a Markov Chain (Gibbs sampler) over elements of an ensemble of mapped weighted networks. Our framework provides insights into the local and global spreading dynamics from arbitrary source nodes and the scaling the of average propagation time for Markovian and non-Markovian processes. Furthermore, it enables efficient source detection and helps to improve strategies for time-critical vaccination. Our framework overcomes the limitations of previous methods such as tree-like assumptions of message passing, omitting dynamical correlations with mean-field approximations, or setting all initial conditions upfront for the kinetic Monte Carlo method. Finally, in a limit of process time, we establish the connection of our mapping with bond percolation.