where · informs about the average over the degree distribution. These quantities tell us that the susceptibility to invasion of a network increases with both the mean and the variance of the degree distribution. This closely echos the results for risk-structured models [10] but with an extra term of −1 due to the network, representing the fact that the route through which an individual acquired infection is closed off for future transmission events. For more structured networks with a local-global distinction, there are two limits in which exact dynamics can also be derived. If the network is composed of m communities of size n 1 , . . . , n m , with the between-community (global) mixing determined by a Poisson process with rate n G and the within-community (local) mixing determined by a Poisson process with rate n L , then in the limit as the communities become large, n i → ∞, the epidemic dynamics on the system areṠ