  By generating the specifics of a network structure only when needed
(on-the-fly), we derive a simple stochastic process that exactly models the
time evolution of susceptible-infectious dynamics on finite-size networks. The
small number of dynamical variables of this birth-death Markov process greatly
simplifies analytical calculations. We show how a dual analytical description,
treating large scale epidemics with a Gaussian approximations and small
outbreaks with a branching process, provides an accurate approximation of the
distribution even for rather small networks. The approach also offers important
computational advantages and generalizes to a vast class of systems.
