We explore a systematic approach to studying the dynamics of evolving networks at a coarse-grained, system level. We emphasize the importance of finding good observables (network properties) in terms of which coarse-grained models can be developed. We illustrate our approach through a particular social network model: the 'rise and fall' of a networked society (Marsili M et al 2004 Proc. Natl Acad. Sci. USA 101 1439). We implement our low-dimensional description computationally using the equation-free approach and show how it can be used to (i) accelerate simulations and (ii) extract system-level stability/bifurcation information from the detailed dynamic model. We discuss other system-level tasks that can be enabled through such a computer-assisted coarse-graining approach.