Time augmented bond percolation mapping of spreading dynamics on networks

In this paper, we propose a mapping of spreading dynamics to weighted networks, where weights represent interaction time delays on edges. With this mapping, we are able to estimate both the process evolution in time and the final outcome of a process. In a limit of process time, we establish the connection of our mapping with the bond percolation and thus we name it time augmented bond percolation mapping. We concentrate on the stochastic formulation of the generalized Susceptible Infected Recovered (SIR) spreading dynamics without memory (exponential inter-event distribution or Poisson process) and with memory (arbitrary inter-event distributions) for arbitrary static network structures including non tree-like networks. Furthermore, we construct a higher-order Markovian representation of process dynamics, where states are time augmented weighted networks and transitions betweeen states are constructed by changing the weight on a randomly selected edge in the time augmented weighted networks. Each weighted network encodes one stochastic outcome of the full process evolution with the same initial conditions and same process parameters. The time augmented bond percolation mapping is constructed in such a way that the time respecting paths (shortest paths) in weighted network preserve the causality of spreading. Moreover, the shortest path distance in weighted networks equals to the physical quantity of propagation time needed for epidemic or information to spread between nodes.