Epidemics on assortatively mixed networks have a faster initial growth rate and a shorter duration than those on disassortatively mixed networks (figure 5a). This is mainly due to the GC containing a 'core' group of high-degree nodes that are highly connected. The differences in the initial epidemic growth rate across the different network types are directly related to the basic reproduction number R 0 (Anderson & May 1991) . The value of R 0 can be estimated as the lead eigenvalue of the next-generation matrix CZ(c ij ) (Diekmann & Heesterbeek 2000) , and, in this case, it can be approximated by the product of the contact matrix and the per-link probability of transmission c ij ZT p M ij . A non-zero entry M ij Z1 denotes that an infectious node j can transmit the infection to a susceptible node i; the magnitude of c ij is given by the per-link transmission probability T p . Estimates of R 0 averaged over 100 generated networks of each type are as follows: R 0 Z12.12T p (s.e. 0.11) for disassortatively mixed networks; R 0 Z15.0T p (s.e. 0.11) for random networks; and R 0 Z17.81T p (s.e. 0.14) for assortatively mixed networks. As expected, the threshold for epidemic outbreaks occurs at lower infection probabilities for assortatively mixed networks than for either disassortatively mixed or random networks.