We study rare events in networks with both internal and external noise, and
develop a general formalism for analyzing rare events that combines
pair-quenched techniques and large-deviation theory. The probability
distribution, shape, and time scale of rare events are considered in detail for
extinction in the Susceptible-Infected-Susceptible model as an illustration. We
find that when both types of noise are present, there is a crossover region as
the network size is increased, where the probability exponent for large
deviations no longer increases linearly with the network size. We demonstrate
that the form of the crossover depends on whether the endemic state is
localized near the epidemic threshold or not
