  We derive a general expression for the probability of global spreading
starting from a single infected seed for contagion processes acting on
generalized, correlated random networks. We employ a simple probabilistic
argument that encodes the spreading mechanism in an intuitive, physical
fashion. We use our approach to directly and systematically obtain triggering
probabilities for contagion processes acting on a collection of random network
families including bipartite random networks. We find the contagion condition,
the location of the phase transition into an endemic state, from an expansion
about the disease-free state.
