Ecological communities have mainly been investigated theoretically in two ways: piecewise, a few species at a time; or as complex networks, simulated in exhaustive detail. But our empirical knowledge of networks is limited, and the space of simulation models and parameters is mindbogglingly vast. We show that a large fraction of that space of possibilities exhibits generic dynamics, which can be predicted from a single minimal model. To demonstrate this, we consider a wide array of ecological models, from resource competition to predation and mutualism, known to display very different behaviors for a few species. We simulate large communities, and show that equilibrium diversity, functioning and stability can often be predicted analytically from only four broad statistical properties of the community. Our approach provides a convenient framework for exploring generic patterns in ecosystem assembly and quantifying the added value of more detailed models and measurements.