Many experiments exploit curvature-driven, surface-diffusion-mediated coarsening for tuning the characteristic structure size of metal network structures made by dealloying, such as nanoporous gold. Here we study this process by kinetic Monte Carlo simulation. The initial microstructures are leveled Gaussian random fields, approximating spinodally decomposed mixtures, of different solid fraction φ. Earlier work establishes these structures as valid representations of the nanoporous gold microstructure. We find that the coarsening law for the characteristic spacing between the ligaments of the network is universal, whereas the time evolution of the characteristic ligament diameter is not. The expected time exponent 1/4 is confirmed by our simulation. Contrary to what may be expected based on continuum models, the degree of surface faceting or roughness has no apparent effect on the coarsening kinetics. In the time interval of our study, the network connectivity - as measured by a scaled density of topological genus - remains sensibly invariant for networks with φ≥0.3, consistent with previous reports of a self-similar evolution of the microstructure during coarsening. Yet, networks with lesser φ lose their connectivity on coarsening and can even undergo a percolation-to-cluster transition. This process is slow for φ only little below 0.3 and it accelerates in networks with lesser φ. The dependency of the connectivity evolution on φ may explain controversial findings on the microstructure evolution of nanoporous gold in experimental studies.