(where the probability of the population being in each possible configuration is calculated) on small, fully connected graphs as summarised in Bailey [126] . The rate at which the complexity of the system of master equations grows means that these equations quickly become too complex to integrate for the most general network. The presence of symmetries in the network, however, does mean that automorphism-driven lumping is one way to manipulate the master equations (whilst preserving the full stochastic information about the system) for solution [127] . At present, this technique has only been applied to relatively simple networks; however, there are no other highly general methods of deriving exact lower-dimensional systems of equations from the master equations. Nevertheless, other specific routes do exist that allow exact systems of equations of lower dimensionality to be derived for special networks. For static networks constructed using the configuration model (where individuals have heterogeneous degree but connections are made at random such that the presence of short loops can be ignored in a large network, see Figure 1 (c)), an exact system of equations for SIR dynamics in the limit of large network size was provided by Ball and Neal [69] . This construction involves attributing to each node an "effective degree", which starts the epidemic at its actual degree, and measures connections still available as routes of infection and is, therefore, reduced by transmission and recovery. Using notation consistent with elsewhere in this paper (and ignoring the global infection terms that were included by Ball and coworkers) this yields the relatively parsimonious set of equationṡ S k = −ρ τ + γ kS k − γ(k + 1)S k+1 ,