  A problem closely related to epidemiology, where a subgraph of 'infected'
links is defined inside a larger network, is investigated. This subgraph is
generated from the underlying network by a random variable, which decides
whether a link is able to propagate a disease/information. The relaxation
timescale of this random variable is examined in both annealed and quenched
limits, and the effectiveness of propagation of disease/information is
analyzed. The dynamics of the model is governed by a master equation and two
types of underlying network are considered: one is scale-free and the other has
exponential degree distribution. We have shown that the relaxation timescale of
the contagion variable has a major influence on the topology of the subgraph of
infected links, which determines the efficiency of spreading of
disease/information over the network.
